GROUPOID SCHEMES 022L Contents 1. Introduction 1 2

Home , Groupoid, Group scheme, Yoneda lemma

022L

Contents 1. Introduction 1 2. Notation 1 3. Equivalence relations 2 4. Group schemes 4 5. Examples of group schemes 5 6. Properties of group schemes 7 7. Properties of group schemes over a field 8 8. Properties of algebraic group schemes 14 9. Abelian varieties 18 10. Actions of group schemes 21 11. Principal homogeneous spaces 22 12. Equivariant quasi-coherent sheaves 23 13. Groupoids 25 14. Quasi-coherent sheaves on groupoids 27 15. Colimits of quasi-coherent modules 29 16. Groupoids and group schemes 34 17. The stabilizer group scheme 34 18. Restricting groupoids 35 19. Invariant subschemes 36 20. Quotient sheaves 38 21. Descent in terms of groupoids 41 22. Separation conditions 42 23. Finite flat groupoids, affine case 43 24. Finite flat groupoids 50 25. Descending quasi-projective schemes 51 26. Other chapters 52 References 54

1. Introduction 022M This chapter is devoted to generalities concerning groupoid schemes. See for example the beautiful paper [KM97] by Keel and Mori.

2. Notation 022N Let S be a scheme. If U, T are schemes over S we denote U(T ) for the set of T -valued points of U over S. In a formula: U(T ) = MorS(T,U). We try to reserve

This is a chapter of the Stacks Project, version fac02ecd, compiled on Sep 14, 2021. 1 GROUPOID SCHEMES 2

the letter T to denote a “test scheme” over S, as in the discussion that follows. Suppose we are given schemes X, Y over S and a morphism of schemes f : X → Y over S. For any scheme T over S we get an induced map of sets f : X(T ) −→ Y (T ) which as indicated we denote by f also. In fact this construction is functorial in the scheme T/S. Yoneda’s Lemma, see Categories, Lemma 3.5, says that f determines and is determined by this transformation of functors f : hX → hY . More generally, we use the same notation for maps between ﬁbre products. For example, if X, Y , Z are schemes over S, and if m : X ×S Y → Z ×S Z is a morphism of schemes over S, then we think of m as corresponding to a collection of maps between T -valued points X(T ) × Y (T ) −→ Z(T ) × Z(T ). And so on and so forth. We continue our convention to label projection maps starting with index 0, so we have pr0 : X ×S Y → X and pr1 : X ×S Y → Y .

3. Equivalence relations 022O Recall that a relation R on a set A is just a subset of R ⊂ A × A. We usually write aRb to indicate (a, b) ∈ R. We say the relation is transitive if aRb, bRc ⇒ aRc. We say the relation is reflexive if aRa for all a ∈ A. We say the relation is symmetric if aRb ⇒ bRa. A relation is called an equivalence relation if it is transitive, reflexive and symmetric. In the setting of schemes we are going to relax the notion of a relation a little bit and just require R → A × A to be a map. Here is the definition. 022PDefinition 3.1. Let S be a scheme. Let U be a scheme over S.

(1) A pre-relation on U over S is any morphism of schemes j : R → U ×S U. In this case we set t = pr0 ◦ j and s = pr1 ◦ j, so that j = (t, s). (2) A relation on U over S is a monomorphism of schemes j : R → U ×S U. (3) A pre-equivalence relation is a pre-relation j : R → U ×S U such that the image of j : R(T ) → U(T ) × U(T ) is an equivalence relation for all T/S. (4) We say a morphism R → U ×S U of schemes is an equivalence relation on U over S if and only if for every scheme T over S the T -valued points of R deﬁne an equivalence relation on the set of T -valued points of U. In other words, an equivalence relation is a pre-equivalence relation such that j is a relation.

02V8 Lemma 3.2. Let S be a scheme. Let U be a scheme over S. Let j : R → U ×S U be a pre-relation. Let g : U 0 → U be a morphism of schemes. Finally, set

0 0 0 0 j 0 0 R = (U ×S U ) ×U×S U R −→ U ×S U Then j0 is a pre-relation on U 0 over S. If j is a relation, then j0 is a relation. If j is a pre-equivalence relation, then j0 is a pre-equivalence relation. If j is an equivalence relation, then j0 is an equivalence relation.

Proof. Omitted. GROUPOID SCHEMES 3

02V9Deﬁnition 3.3. Let S be a scheme. Let U be a scheme over S. Let j : R → U ×S U be a pre-relation. Let g : U 0 → U be a morphism of schemes. The pre-relation 0 0 0 0 0 j : R → U ×S U is called the restriction, or pullback of the pre-relation j to U . 0 In this situation we sometimes write R = R|U 0 .

022Q Lemma 3.4. Let j : R → U ×S U be a pre-relation. Consider the relation on points of the scheme U defined by the rule x ∼ y ⇔ ∃ r ∈ R : t(r) = x, s(r) = y. If j is a pre-equivalence relation then this is an equivalence relation. Proof. Suppose that x ∼ y and y ∼ z. Pick r ∈ R with t(r) = x, s(r) = y and pick r0 ∈ R with t(r0) = y, s(r0) = z. Pick a field K fitting into the following commutative diagram κ(r) / K O O

κ(y) / κ(r0)

Denote xK , yK , zK : Spec(K) → U the morphisms Spec(K) → Spec(κ(r)) → Spec(κ(x)) → U Spec(K) → Spec(κ(r)) → Spec(κ(y)) → U Spec(K) → Spec(κ(r0)) → Spec(κ(z)) → U

By construction (xK , yK ) ∈ j(R(K)) and (yK , zK ) ∈ j(R(K)). Since j is a pre- equivalence relation we see that also (xK , zK ) ∈ j(R(K)). This clearly implies that x ∼ z.

The proof that ∼ is reﬂexive and symmetric is omitted.

0DT7 Lemma 3.5. Let j : R → U ×S U be a pre-relation. Assume (1) s, t are unramiﬁed, (2) for any algebraically closed ﬁeld k over S the map R(k) → U(k) × U(k) is an equivalence relation, (3) there are morphisms e : U → R, i : R → R, c : R ×s,U,t R → R such that

U / R R / R R ×s,U,t R / R e i c

∆ j j j j×j j

flip pr02 U ×S U / U ×S UU ×S U / U ×S UU ×S U ×S U / U ×S U are commutative. Then j is an equivalence relation. Proof. By condition (1) and Morphisms, Lemma 35.16 we see that j is a unram-

iﬁed. Then ∆j : R → R ×U×S U R is an open immersion by Morphisms, Lemma 35.13. However, then condition (2) says ∆j is bijective on k-valued points, hence ∆j is an isomorphism, hence j is a monomorphism. Then it easily follows from the commutative diagrams that R(T ) ⊂ U(T ) × U(T ) is an equivalence relation for all schemes T over S. GROUPOID SCHEMES 4

4. Group schemes 022R Let us recall that a group is a pair (G, m) where G is a set, and m : G × G → G is a map of sets with the following properties: (1) (associativity) m(g, m(g0, g00)) = m(m(g, g0), g00) for all g, g0, g00 ∈ G, (2) (identity) there exists a unique element e ∈ G (called the identity, unit, or 1 of G) such that m(g, e) = m(e, g) = g for all g ∈ G, and (3) (inverse) for all g ∈ G there exists a i(g) ∈ G such that m(g, i(g)) = m(i(g), g) = e, where e is the identity. Thus we obtain a map e : {∗} → G and a map i : G → G so that the quadruple (G, m, e, i) satisfies the axioms listed above. A homomorphism of groups ψ :(G, m) → (G0, m0) is a map of sets ψ : G → G0 such that m0(ψ(g), ψ(g0)) = ψ(m(g, g0)). This automatically insures that ψ(e) = e0 and i0(ψ(g)) = ψ(i(g)). (Obvious notation.) We will use this below. 022SDefinition 4.1. Let S be a scheme. (1) A group scheme over S is a pair (G, m), where G is a scheme over S and m : G ×S G → G is a morphism of schemes over S with the following property: For every scheme T over S the pair (G(T ), m) is a group. (2) A morphism ψ :(G, m) → (G0, m0) of group schemes over S is a morphism ψ : G → G0 of schemes over S such that for every T/S the induced map ψ : G(T ) → G0(T ) is a homomorphism of groups. Let (G, m) be a group scheme over the scheme S. By the discussion above (and the discussion in Section 2) we obtain morphisms of schemes over S: (identity) e : S → G and (inverse) i : G → G such that for every T the quadruple (G(T ), m, e, i) satisfies the axioms of a group listed above. Let (G, m), (G0, m0) be group schemes over S. Let f : G → G0 be a morphism of schemes over S. It follows from the definition that f is a morphism of group schemes over S if and only if the following diagram is commutative: 0 0 G ×S G / G ×S G f×f m m f G / G0 022T Lemma 4.2. Let (G, m) be a group scheme over S. Let S0 → S be a morphism 0 of schemes. The pullback (GS0 , mS0 ) is a group scheme over S . Proof. Omitted. 047DDefinition 4.3. Let S be a scheme. Let (G, m) be a group scheme over S. (1) A closed subgroup scheme of G is a closed subscheme H ⊂ G such that

m|H×S H factors through H and induces a group scheme structure on H over S. (2) An open subgroup scheme of G is an open subscheme G0 ⊂ G such that 0 0 0 0 m|G ×S G factors through G and induces a group scheme structure on G over S. Alternatively, we could say that H is a closed subgroup scheme of G if it is a group scheme over S endowed with a morphism of group schemes i : H → G over S which identiﬁes H with a closed subscheme of G. GROUPOID SCHEMES 5

0G8L Lemma 4.4. Let S be a scheme. Let (G, m, e, i) be a group scheme over S. (1) A closed subscheme H ⊂ G is a closed subgroup scheme if and only if

e : S → G, m|H×S H : H ×S H → G, and i|H : H → G factor through H. (2) An open subscheme H ⊂ G is an open subgroup scheme if and only if

e : S → G, m|H×S H : H ×S H → G, and i|H : H → G factor through H. Proof. Looking at T -valued points this translates into the well known conditions characterizing subsets of groups as subgroups. 047EDefinition 4.5. Let S be a scheme. Let (G, m) be a group scheme over S. (1) We say G is a smooth group scheme if the structure morphism G → S is smooth. (2) We say G is a flat group scheme if the structure morphism G → S is flat. (3) We say G is a separated group scheme if the structure morphism G → S is separated. Add more as needed.

5. Examples of group schemes 047F 022UExample 5.1 (Multiplicative group scheme). Consider the functor which asso- ∗ ciates to any scheme T the group Γ(T, OT ) of units in the global sections of the structure sheaf. This is representable by the scheme −1 Gm = Spec(Z[x, x ]) The morphism giving the group structure is the morphism

Gm × Gm → Gm −1 −1 −1 Spec(Z[x, x ] ⊗Z Z[x, x ]) → Spec(Z[x, x ]) −1 −1 −1 Z[x, x ] ⊗Z Z[x, x ] ← Z[x, x ] x ⊗ x ← x

Hence we see that Gm is a group scheme over Z. For any scheme S the base change Gm,S is a group scheme over S whose functor of points is ∗ T/S 7−→ Gm,S(T ) = Gm(T ) = Γ(T, OT ) as before. 040MExample 5.2 (Roots of unity). Let n ∈ N. Consider the functor which associates ∗ to any scheme T the subgroup of Γ(T, OT ) consisting of nth roots of unity. This is representable by the scheme n µn = Spec(Z[x]/(x − 1)). The morphism giving the group structure is the morphism

µn × µn → µn n n n Spec(Z[x]/(x − 1) ⊗Z Z[x]/(x − 1)) → Spec(Z[x]/(x − 1)) n n n Z[x]/(x − 1) ⊗Z Z[x]/(x − 1) ← Z[x]/(x − 1) x ⊗ x ← x GROUPOID SCHEMES 6

Hence we see that µn is a group scheme over Z. For any scheme S the base change µn,S is a group scheme over S whose functor of points is ∗ n T/S 7−→ µn,S(T ) = µn(T ) = {f ∈ Γ(T, OT ) | f = 1} as before. 022VExample 5.3 (Additive group scheme). Consider the functor which associates to any scheme T the group Γ(T, OT ) of global sections of the structure sheaf. This is representable by the scheme

Ga = Spec(Z[x]) The morphism giving the group structure is the morphism

Ga × Ga → Ga

Spec(Z[x] ⊗Z Z[x]) → Spec(Z[x])

Z[x] ⊗Z Z[x] ← Z[x] x ⊗ 1 + 1 ⊗ x ← x

Hence we see that Ga is a group scheme over Z. For any scheme S the base change Ga,S is a group scheme over S whose functor of points is

T/S 7−→ Ga,S(T ) = Ga(T ) = Γ(T, OT ) as before. 022WExample 5.4 (General linear group scheme). Let n ≥ 1. Consider the functor which associates to any scheme T the group

GLn(Γ(T, OT )) of invertible n × n matrices over the global sections of the structure sheaf. This is representable by the scheme

GLn = Spec(Z[{xij}1≤i,j≤n][1/d])

where d = det((xij)) with (xij) the n × n matrix with entry xij in the (i, j)-spot. The morphism giving the group structure is the morphism

GLn × GLn → GLn

Spec(Z[xij, 1/d] ⊗Z Z[xij, 1/d]) → Spec(Z[xij, 1/d])

Z[xij, 1/d] ⊗Z Z[xij, 1/d] ← Z[xij, 1/d] X xik ⊗ xkj ← xij

Hence we see that GLn is a group scheme over Z. For any scheme S the base change GLn,S is a group scheme over S whose functor of points is

T/S 7−→ GLn,S(T ) = GLn(T ) = GLn(Γ(T, OT )) as before. 022XExample 5.5. The determinant deﬁnes a morphism of group schemes

det : GLn −→ Gm

over Z. By base change it gives a morphism of group schemes GLn,S → Gm,S over any base scheme S. GROUPOID SCHEMES 7

03YWExample 5.6 (Constant group). Let G be an abstract group. Consider the functor which associates to any scheme T the group of locally constant maps T → G (where T has the Zariski topology and G the discrete topology). This is representable by the scheme a GSpec(Z) = Spec(Z). g∈G The morphism giving the group structure is the morphism

GSpec(Z) ×Spec(Z) GSpec(Z) −→ GSpec(Z) which maps the component corresponding to the pair (g, g0) to the component 0 corresponding to gg . For any scheme S the base change GS is a group scheme over S whose functor of points is

T/S 7−→ GS(T ) = {f : T → G locally constant} as before.

6. Properties of group schemes 045W In this section we collect some simple properties of group schemes which hold over any base. 047G Lemma 6.1. Let S be a scheme. Let G be a group scheme over S. Then G → S is separated (resp. quasi-separated) if and only if the identity morphism e : S → G is a closed immersion (resp. quasi-compact). Proof. We recall that by Schemes, Lemma 21.11 we have that e is an immersion which is a closed immersion (resp. quasi-compact) if G → S is separated (resp. quasi-separated). For the converse, consider the diagram

G / G ×S G ∆G/S (g,g0)7→m(i(g),g0) e S / G It is an exercise in the functorial point of view in algebraic geometry to show that this diagram is cartesian. In other words, we see that ∆G/S is a base change of e. Hence if e is a closed immersion (resp. quasi-compact) so is ∆G/S, see Schemes, Lemma 18.2 (resp. Schemes, Lemma 19.3). 047H Lemma 6.2. Let S be a scheme. Let G be a group scheme over S. Let T be a scheme over S and let ψ : T → G be a morphism over S. If T is ﬂat over S, then the morphism T ×S G −→ G, (t, g) 7−→ m(ψ(t), g)

is flat. In particular, if G is flat over S, then m : G ×S G → G is flat. Proof. Consider the diagram

T ×S G / T ×S G / G (t,g)7→(t,m(ψ(t),g)) pr

T / S The left top horizontal arrow is an isomorphism and the square is cartesian. Hence the lemma follows from Morphisms, Lemma 25.8. GROUPOID SCHEMES 8

047I Lemma 6.3. Let (G, m, e, i) be a group scheme over the scheme S. Denote f : [MvdGE, G → S the structure morphism. Then there exist canonical isomorphisms Proposition 3.15] ∼ ∗ ∼ ∗ ∗ ΩG/S = f CS/G = f e ΩG/S

where CS/G denotes the conormal sheaf of the immersion e. In particular, if S is the spectrum of a ﬁeld, then ΩG/S is a free OG-module. Proof. By Morphisms, Lemma 32.10 we have

∗ ΩG×S G/G = pr0ΩG/S

where on the left hand side we view G ×S G as a scheme over G using pr1. Let τ : G ×S G → G ×S G be the “shearing map” given by (g, h) 7→ (m(g, h), h) on points. This map is an automorphism of G ×S G viewed as a scheme over G via the projection pr1. Combining these two remarks we obtain an isomorphism ∗ ∗ ∗ τ pr0ΩG/S → pr0ΩG/S

Since pr0 ◦ τ = m this can be rewritten as an isomorphism ∗ ∗ m ΩG/S → pr0ΩG/S

Pulling back this isomorphism by (e ◦ f, idG): G → G ×S G and using that m ◦ (e ◦ f, idG) = idG and pr0 ◦ (e ◦ f, idG) = e ◦ f we obtain an isomorphism ∗ ∗ ΩG/S → f e ΩG/S ∼ ∗ as desired. By Morphisms, Lemma 32.16 we have CS/G = e ΩG/S. If S is the spectrum of a ﬁeld, then any OS-module on S is free and the ﬁnal statement follows.

0BF5 Lemma 6.4. Let S be a scheme. Let G be a group scheme over S. Let s ∈ S. Then the composition

TG/S,e(s) ⊕ TG/S,e(s) = TG×S G/S,(e(s),e(s)) → TG/S,e(s) is addition of tangent vectors. Here the = comes from Varieties, Lemma 16.7 and the right arrow is induced from m : G ×S G → G via Varieties, Lemma 16.6. Proof. We will use Varieties, Equation (16.3.1) and work with tangent vectors

in ﬁbres. An element θ in the ﬁrst factor TGs/s,e(s) is the image of θ via the

map TGs/s,e(s) → TGs×Gs/s,(e(s),e(s)) coming from (1, e): Gs → Gs × Gs. Since m ◦ (1, e) = 1 we see that θ maps to θ by functoriality. Since the map is linear we see that (θ1, θ2) maps to θ1 + θ2.

7. Properties of group schemes over a field 047J In this section we collect some properties of group schemes over a field. In the case of group schemes which are (locally) algebraic over a field we can say a lot more, see Section 8.

047K Lemma 7.1. If (G, m) is a group scheme over a ﬁeld k, then the multiplication map m : G ×k G → G is open. GROUPOID SCHEMES 9

Proof. The multiplication map is isomorphic to the projection map pr0 : G×k G → G because the diagram

G ×k G / G ×k G (g,g0)7→(m(g,g0),g0) m (g,g0)7→g id G / G is commutative with isomorphisms as horizontal arrows. The projection is open by Morphisms, Lemma 23.4. 0B7N Lemma 7.2. If (G, m) is a group scheme over a ﬁeld k. Let U ⊂ G open and T → G a morphism of schemes. Then the image of the composition T ×k U → G ×k G → G is open.

Proof. For any field extension k ⊂ K the morphism GK → G is open (Morphisms, Lemma 23.4). Every point ξ of T ×kU is the image of a morphism (t, u) : Spec(K) → T ×k U for some K. Then the image of TK ×K UK = (T ×k U)K → GK contains the translate t · UK which is open. Combining these facts we see that the image of T ×k U → G contains an open neighbourhood of the image of ξ. Since ξ was arbitrary we win. 047L Lemma 7.3. Let G be a group scheme over a field. Then G is a separated scheme. Proof. Say S = Spec(k) with k a field, and let G be a group scheme over S. By Lemma 6.1 we have to show that e : S → G is a closed immersion. By Morphisms, Lemma 20.2 the image of e : S → G is a closed point of G. It is clear that OG → e∗OS is surjective, since e∗OS is a skyscraper sheaf supported at the neutral element of G with value k. We conclude that e is a closed immersion by Schemes, Lemma 24.2. 047M Lemma 7.4. Let G be a group scheme over a field k. Then

(1) every local ring OG,g of G has a unique minimal prime ideal, (2) there is exactly one irreducible component Z of G passing through e, and (3) Z is geometrically irreducible over k. Proof. For any point g ∈ G there exists a ﬁeld extension k ⊂ K and a K-valued point g0 ∈ G(K) mapping to g. If we think of g0 as a K-rational point of the group

0 scheme GK , then we see that OG,g → OGK ,g is a faithfully flat local ring map (as GK → G is flat, and a local flat ring map is faithfully flat, see Algebra, Lemma 0 39.17). The result for OGK ,g implies the result for OG,g, see Algebra, Lemma 30.5. Hence in order to prove (1) it suffices to prove it for k-rational points g of G. In this case translation by g defines an automorphism G → G which maps e to g. Hence ∼ OG,g = OG,e. In this way we see that (2) implies (1), since irreducible components passing through e correspond one to one with minimal prime ideals of OG,e. In order to prove (2) and (3) it suffices to prove (2) when k is algebraically closed. In this case, let Z1, Z2 be two irreducible components of G passing through e. Since k is algebraically closed the closed subscheme Z1 ×k Z2 ⊂ G ×k G is irreducible too, see Varieties, Lemma 8.4. Hence m(Z1 ×k Z2) is contained in an irreducible component of G. On the other hand it contains Z1 and Z2 since m|e×G = idG and m|G×e = idG. We conclude Z1 = Z2 as desired. GROUPOID SCHEMES 10

04L9Remark 7.5. Warning: The result of Lemma 7.4 does not mean that every irreducible component of G/k is geometrically irreducible. For example the group 3 scheme µ3,Q = Spec(Q[x]/(x − 1)) over Q has two irreducible components corresponding to the factorization x3 − 1 = (x − 1)(x2 + x + 1). The ﬁrst factor corresponds to the irreducible component passing through the identity, and the second irreducible component is not geometrically irreducible over Spec(Q). 047R Lemma 7.6. Let G be a group scheme over a perfect ﬁeld k. Then the reduction Gred of G is a closed subgroup scheme of G.

Proof. Omitted. Hint: Use that Gred ×k Gred is reduced by Varieties, Lemmas 6.3 and 6.7. 047S Lemma 7.7. Let k be a field. Let ψ : G0 → G be a morphism of group schemes over k. If ψ(G0) is open in G, then ψ(G0) is closed in G. Proof. Let U = ψ(G0) ⊂ G. Let Z = G \ ψ(G0) = G \ U with the reduced induced closed subscheme structure. By Lemma 7.2 the image of 0 Z ×k G −→ Z ×k U −→ G is open (the first arrow is surjective). On the other hand, since ψ is a homomorphism 0 of group schemes, the image of Z ×k G → G is contained in Z (because translation by ψ(g0) preserves U for all points g0 of G0; small detail omitted). Hence Z ⊂ G is an open subset (although not necessarily an open subscheme). Thus U = ψ(G0) is closed. 047T Lemma 7.8. Let i : G0 → G be an immersion of group schemes over a field k. Then i is a closed immersion, i.e., i(G0) is a closed subgroup scheme of G. Proof. To show that i is a closed immersion it suffices to show that i(G0) is a 0 0 closed subset of G. Let k ⊂ k be a perfect extension of k. If i(Gk0 ) ⊂ Gk0 is 0 closed, then i(G ) ⊂ G is closed by Morphisms, Lemma 25.12 (as Gk0 → G is flat, quasi-compact and surjective). Hence we may and do assume k is perfect. We will use without further mention that products of reduced schemes over k are reduced. We may replace G0 and G by their reductions, see Lemma 7.6. Let G0 ⊂ G be the closure of i(G0) viewed as a reduced closed subscheme. By Varieties, Lemma 24.1 0 0 0 0 we conclude that G ×k G is the closure of the image of G ×k G → G ×k G. Hence 0 0 0 m G ×k G ⊂ G as m is continuous. It follows that G0 ⊂ G is a (reduced) closed subgroup scheme. By Lemma 7.7 we see that i(G0) ⊂ G0 is also closed which implies that i(G0) = G0 as desired. 0B7P Lemma 7.9. Let G be a group scheme over a field k. If G is irreducible, then G is quasi-compact.

Proof. Suppose that k ⊂ K is a field extension. If GK is quasi-compact, then G is too as GK → G is surjective. By Lemma 7.4 we see that GK is irreducible. Hence it suffices to prove the lemma after replacing k by some extension. Choose K to be an algebraically closed field extension of very large cardinality. Then by Varieties, Lemma 14.2, we see that GK is a Jacobson scheme all of whose closed points have residue field equal to K. In other words we may assume G is a Jacobson scheme all of whose closed points have residue field k. GROUPOID SCHEMES 11

Let U ⊂ G be a nonempty affine open. Let g ∈ G(k). Then gU ∩ U 6= ∅. Hence we see that g is in the image of the morphism −1 U ×Spec(k) U −→ G, (u1, u2) 7−→ u1u2 Since the image of this morphism is open (Lemma 7.1) we see that the image is all of G (because G is Jacobson and closed points are k-rational). Since U is affine, so is U ×Spec(k) U. Hence G is the image of a quasi-compact scheme, hence quasi-compact. 0B7Q Lemma 7.10. Let G be a group scheme over a field k. If G is connected, then G is irreducible. Proof. By Varieties, Lemma 7.13 we see that G is geometrically connected. If we show that GK is irreducible for some field extension k ⊂ K, then the lemma follows. Hence we may apply Varieties, Lemma 14.2 to reduce to the case where k is algebraically closed, G is a Jacobson scheme, and all the closed points are k-rational. Let Z ⊂ G be the unique irreducible component of G passing through the neutral element, see Lemma 7.4. Endowing Z with the reduced induced closed subscheme structure, we see that Z ×k Z is reduced and irreducible (Varieties, Lemmas 6.7

and 8.4). We conclude that m|Z×kZ : Z ×k Z → G factors through Z. Hence Z becomes a closed subgroup scheme of G. To get a contradiction, assume there exists another irreducible component Z0 ⊂ G. Then Z ∩ Z0 = ∅ by Lemma 7.4. By Lemma 7.9 we see that Z is quasi-compact. Thus we may choose a quasi-compact open U ⊂ G with Z ⊂ U and U ∩ Z0 = ∅. The image W of Z ×k U → G is open in G by Lemma 7.2. On the other hand, W is quasi-compact as the image of a quasi-compact space. We claim that W is closed. If the claim is true, then W ⊂ G \ Z0 is a proper open and closed subset of G, which contradicts the assumption that G is connected. Proof of the claim. Since W is quasi-compact, we see that points in the closure of W are specializations of points of W (Morphisms, Lemma 6.5). Thus we have to show that any irreducible component Z00 ⊂ G of G which meets W is contained in W . As G is Jacobson and closed points are rational, Z00 ∩ W has a rational point 00 00 g ∈ Z (k) ∩ W (k) and hence Z = Zg. But W = m(Z ×k W ) by construction, so 00 00 Z ∩ W 6= ∅ implies Z ⊂ W . 0B7R Proposition 7.11. Let G be a group scheme over a field k. There exists a canonical closed subgroup scheme G0 ⊂ G with the following properties (1) G0 → G is a flat closed immersion, (2) G0 ⊂ G is the connected component of the identity, (3) G0 is geometrically irreducible, and (4) G0 is quasi-compact. Proof. Let G0 be the connected component of the identity with its canonical scheme structure (Morphisms, Definition 26.3). To show that G0 is a closed subs- group scheme we will use the criterion of Lemma 4.4. The morphism e : Spec(k) → G factors through G0 as we chose G0 to be the connected component of G containing e. Since i : G → G is an automorphism fixing e, we see that i sends G0 into itself. By Varieties, Lemma 7.13 the scheme G0 is geometrically connected over k. GROUPOID SCHEMES 12

0 0 0 0 0 Thus G ×k G is connected (Varieties, Lemma 7.4). Thus m(G ×k G ) ⊂ G set 0 0 0 0 0 theoretically. Thus m|G ×kG : G ×k G → G factors through G by Morphisms, Lemma 26.1. Hence G0 is a closed subgroup scheme of G. By Lemma 7.10 we see that G0 is irreducible. By Lemma 7.4 we see that G0 is geometrically irreducible. 0 By Lemma 7.9 we see that G is quasi-compact. 0B7T Lemma 7.12. Let k be a field. Let T = Spec(A) where A is a directed colimit of algebras which are finite products of copies of k. For any scheme X over k we have |T ×k X| = |T | × |X| as topological spaces. Proof. By taking an affine open covering we reduce to the case of an affine X. Say X = Spec(B). Write A = colim A with A = Q k and T finite. Then i i t∈Ti i Ti = | Spec(Ai)| with the discrete topology and the transition morphisms Ai → Ai0 are given by set maps Ti0 → Ti. Thus |T | = lim Ti as a topological space, see Limits, Lemma 4.6. Similarly we have

|T ×k X| = | Spec(A ⊗k B)|

= | Spec(colim Ai ⊗k B)|

= (lim Ti) × |X| = |T | × |X| by the lemma above and the fact that limits commute with limits. The following lemma says that in fact we can put a “algebraic profinite family of points” in an affine open. We urge the reader to read Lemma 8.6 first. 0B7U Lemma 7.13. Let k be an algebraically closed field. Let G be a group scheme over k. Assume that G is Jacobson and that all closed points are k-rational. Let T = Spec(A) where A is a directed colimit of algebras which are finite products of copies of k. For any morphism f : T → G there exists an affine open U ⊂ G containing f(T ). Proof. Let G0 ⊂ G be the closed subgroup scheme found in Proposition 7.11. The first two paragraphs serve to reduce to the case G = G0. Observe that T is a directed inverse limit of finite topological spaces (Limits, Lemma 4.6), hence profinite as a topological space (Topology, Definition 22.1). Let W ⊂ G be a quasi-compact open containing the image of T → G. After replacing W by the image of G0 × W → G × G → G we may assume that W is invariant under the action of left translation by G0, see Lemma 7.2. Consider the composition f π ψ = π ◦ f : T −→ W −→ π0(W )

The space π0(W ) is profinite (Topology, Lemma 23.9 and Properties, Lemma 2.4). Let Fξ ⊂ T be the fibre of T → π0(W ) over ξ ∈ π0(W ). Assume that for all ξ we can find an affine open Uξ ⊂ W with F ⊂ U. Since ψ : T → π0(W ) is proper as a map of topological spaces (Topology, Lemma 17.7), we can find a quasi-compact open −1 −1 Vξ ⊂ π0(W ) such that ψ (Vξ) ⊂ f (Uξ) (easy topological argument omitted). −1 After replacing Uξ by Uξ ∩ π (Vξ), which is open and closed in Uξ hence affine, we GROUPOID SCHEMES 13

−1 −1 see that Uξ ⊂ π (Vξ) and Uξ ∩ T = ψ (Vξ). By Topology, Lemma 22.4 we can S ﬁnd a ﬁnite disjoint union decomposition π0(W ) = i=1,...,n Vi by quasi-compact

opens such that Vi ⊂ Vξi for some i. Then we see that [ −1 f(T ) ⊂ Uξ ∩ π (Vi) i=1,...,n i the right hand side of which is a finite disjoint union of affines, therefore affine. Let Z be a connected component of G which meets f(T ). Then Z has a k-rational point z (because all residue fields of the scheme T are isomorphic to k). Hence Z = G0z. By our choice of W , we see that Z ⊂ W . The argument in the preceding paragraph reduces us to the problem of finding an affine open neighbourhood of f(T ) ∩ Z in W . After translation by a rational point we may assume that Z = G0 (details omitted). Observe that the scheme theoretic inverse image T 0 = f −1(G0) ⊂ T is a closed subscheme, which has the same type. After replacing T by T 0 we may assume that f(T ) ⊂ G0. Choose an affine open neighbourhood U ⊂ G of e ∈ G, so that in particular U ∩ G0 is nonempty. We will show there exists a g ∈ G0(k) such that f(T ) ⊂ g−1U. This will finish the proof as g−1U ⊂ W by the left G0-invariance of W . The arguments in the preceding two paragraphs allow us to pass to G0 and reduce the problem to the following: Assume G is irreducible and U ⊂ G an affine open neighbourhood of e. Show that f(T ) ⊂ g−1U for some g ∈ G(k). Consider the morphism −1 U ×k T −→ G ×k T, (t, u) −→ (uf(t) , t)

which is an open immersion (because the extension of this morphism to G ×k T → G×k T is an isomorphism). By our assumption on T we see that we have |U ×k T | = |U|×|T | and similarly for G×k T , see Lemma 7.12. Hence the image of the displayed S open immersion is a finite union of boxes i=1,...,n Ui × Vi with Vi ⊂ T and Ui ⊂ G quasi-compact open. This means that the possible opens Uf(t)−1, t ∈ T are finite −1 −1 in number, say Uf(t1) , . . . , Uf(tr) . Since G is irreducible the intersection −1 −1 Uf(t1) ∩ ... ∩ Uf(tr) is nonempty and since G is Jacobson with closed points k-rational, we can choose a k-valued point g ∈ G(k) of this intersection. Then we see that g ∈ Uf(t)−1 for −1 all t ∈ T which means that f(t) ∈ g U as desired. 047VRemark 7.14. If G is a group scheme over a field, is there always a quasi- compact open and closed subgroup scheme? By Proposition 7.11 this question is only interesting if G has infinitely many connected components (geometrically). 047U Lemma 7.15. Let G be a group scheme over a field. There exists an open and closed subscheme G0 ⊂ G which is a countable union of affines. Proof. Let e ∈ U(k) be a quasi-compact open neighbourhood of the identity element. By replacing U by U ∩ i(U) we may assume that U is invariant under the inverse map. As G is separated this is still a quasi-compact set. Set

0 [ G = mn(U ×k ... ×k U) n≥1

where mn : G ×k ... ×k G → G is the n-slot multiplication map (g1, . . . , gn) 7→ m(m(... (m(g1, g2), g3),...), gn). Each of these maps are open (see Lemma 7.1) GROUPOID SCHEMES 14

hence G0 is an open subgroup scheme. By Lemma 7.7 it is also a closed subgroup scheme. 8. Properties of algebraic group schemes 0BF6 Recall that a scheme over a field k is (locally) algebraic if it is (locally) of finite type over Spec(k), see Varieties, Definition 20.1. This is the sense of algebraic we are using in the title of this section. 045X Lemma 8.1. Let k be a field. Let G be a locally algebraic group scheme over k. Then G is equidimensional and dim(G) = dimg(G) for all g ∈ G. For any closed point g ∈ G we have dim(G) = dim(OG,g). 0 Proof. Let us first prove that dimg(G) = dimg0 (G) for any pair of points g, g ∈ G. By Morphisms, Lemma 28.3 we may extend the ground field at will. Hence we may assume that both g and g0 are defined over k. Hence there exists an automorphism of G mapping g to g0, whence the equality. By Morphisms, Lemma 28.1 we have dimg(G) = dim(OG,g) + trdegk(κ(g)). On the other hand, the dimension of G (or any open subset of G) is the supremum of the dimensions of the local rings of G, see Properties, Lemma 10.3. Clearly this is maximal for closed points g in which case trdegk(κ(g)) = 0 (by the Hilbert Nullstellensatz, see Morphisms, Section 16). Hence the lemma follows. The following result is sometimes referred to as Cartier’s theorem. 047N Lemma 8.2. Let k be a field of characteristic 0. Let G be a locally algebraic group scheme over k. Then the structure morphism G → Spec(k) is smooth, i.e., G is a smooth group scheme. Proof. By Lemma 6.3 the module of differentials of G over k is free. Hence smoothness follows from Varieties, Lemma 25.1. 047ORemark 8.3. Any group scheme over a field of characteristic 0 is reduced, see [Per75, I, Theorem 1.1 and I, Corollary 3.9, and II, Theorem 2.4] and also [Per76, Proposition 4.2.8]. This was a question raised in [Oor66, page 80]. We have seen in Lemma 8.2 that this holds when the group scheme is locally of finite type. 047P Lemma 8.4. Let k be a perfect field of characteristic p > 0 (see Lemma 8.2 for the characteristic zero case). Let G be a locally algebraic group scheme over k. If G is reduced then the structure morphism G → Spec(k) is smooth, i.e., G is a smooth group scheme.

Proof. By Lemma 6.3 the sheaf ΩG/k is free. Hence the lemma follows from Varieties, Lemma 25.2. 047QRemark 8.5. Let k be a field of characteristic p > 0. Let α ∈ k be an element which is not a pth power. The closed subgroup scheme p p 2 G = V (x + αy ) ⊂ Ga,k is reduced and irreducible but not smooth (not even normal). The following lemma is a special case of Lemma 7.13 with a somewhat easier proof. 0B7S Lemma 8.6. Let k be an algebraically closed field. Let G be a locally algebraic group scheme over k. Let g1, . . . , gn ∈ G(k) be k-rational points. Then there exists an affine open U ⊂ G containing g1, . . . , gn. GROUPOID SCHEMES 15

Proof. We first argue by induction on n that we may assume all gi are on the same connected component of G. Namely, if not, then we can find a decomposition G = W1 q W2 with Wi open in G and (after possibly renumbering) g1, . . . , gr ∈ W1 and gr+1, . . . , gn ∈ W2 for some 0 < r < n. By induction we can find affine opens U1 and U2 of G with g1, . . . , gr ∈ U1 and gr+1, . . . , gn ∈ U2. Then

g1, . . . , gn ∈ (U1 ∩ W1) ∪ (U2 ∩ W2)

is a solution to the problem. Thus we may assume g1, . . . , gn are all on the same −1 0 connected component of G. Translating by g1 we may assume g1, . . . , gn ∈ G where G0 ⊂ G is as in Proposition 7.11. Choose an aﬃne open neighbourhood U of e, in particular U ∩ G0 is nonempty. Since G0 is irreducible we see that

0 −1 −1 G ∩ (Ug1 ∩ ... ∩ Ugn )

is nonempty. Since G → Spec(k) is locally of finite type, also G0 → Spec(k) is locally of finite type, hence any nonempty open has a k-rational point. Thus we 0 −1 −1 −1 can pick g ∈ G (k) with g ∈ Ugi for all i. Then gi ∈ g U for all i and g U is the affine open we were looking for.

0BF7 Lemma 8.7. Let k be a ﬁeld. Let G be an algebraic group scheme over k. Then G is quasi-projective over k.

Proof. By Varieties, Lemma 15.1 we may assume that k is algebraically closed. Let G0 ⊂ G be the connected component of G as in Proposition 7.11. Then every other connected component of G has a k-rational point and hence is isomorphic to G0 as a scheme. Since G is quasi-compact and Noetherian, there are ﬁnitely many of these connected components. Thus we reduce to the case discussed in the next paragraph.

Let G be a connected algebraic group scheme over an algebraically closed ﬁeld k. If the characteristic of k is zero, then G is smooth over k by Lemma 8.2. If the characteristic of k is p > 0, then we let H = Gred be the reduction of G. By Divisors, Proposition 17.9 it suﬃces to show that H has an ample invertible sheaf. (For an algebraic scheme over k having an ample invertible sheaf is equivalent to being quasi-projective over k, see for example the very general More on Morphisms, Lemma 45.1.) By Lemma 7.6 we see that H is a group scheme over k. By Lemma 8.4 we see that H is smooth over k. This reduces us to the situation discussed in the next paragraph.

Let G be a quasi-compact irreducible smooth group scheme over an algebraically closed field k. Observe that the local rings of G are regular and hence UFDs (Varieties, Lemma 25.3 and More on Algebra, Lemma 118.2). The complement of a nonempty affine open of G is the support of an effective Cartier divisor D. This follows from Divisors, Lemma 16.6. (Observe that G is separated by Lemma 7.3.) We conclude there exists an effective Cartier divisor D ⊂ G such that G \ D is affine. We will use below that for any n ≥ 1 and g1, . . . , gn ∈ G(k) the complement S ∼ G\ Dgi is affine. Namely, it is the intersection of the affine opens G\Dgi = G\D in the separated scheme G. GROUPOID SCHEMES 16

We may choose the top row of the diagram

j π d GUo / Ak O O

π0 W / V such that U 6= ∅, j : U → G is an open immersion, and π is étale, see Morphisms, d Lemma 36.20. There is a nonempty affine open V ⊂ Ak such that with W = −1 0 0 π (V ) the morphism π = π|W : W → V is finite étale. In particular π is finite locally free, say of degree n. Consider the effective Cartier divisor D = {(g, w) | m(g, j(w)) ∈ D} ⊂ G × W (This is the restriction to G × W of the pullback of D ⊂ G under the flat morphism m : G × G → G.) Consider the closed subset1 T = (1 × π0)(D) ⊂ G × V . Since π0 is finite locally free, every irreducible component of T has codimension 1 in G × V . Since G × V is smooth over k we conclude these components are effective Cartier divisors (Divisors, Lemma 15.7 and lemmas cited above) and hence T is the support of an effective Cartier divisor E in G × V . If v ∈ V (k), then (π0)−1(v) = {w1, . . . , wn} ⊂ W (k) and we see that [ −1 Ev = Dj(wi) i=1,...,n

in G set theoretically. In particular we see that G \ Ev is affine open (see above). Moreover, if g ∈ G(k), then there exists a v ∈ V such that g 6∈ Ev. Namely, the set W 0 of w ∈ W such that g 6∈ Dj(w)−1 is nonempty open and it suffices to pick v such that the fibre of W 0 → V over v has n elements.

Consider the invertible sheaf M = OG×V (E) on G × V . By Varieties, Lemma 30.5 the isomorphism class L of the restriction Mv = OG(Ev) is independent of v ∈ V (k). On the other hand, for every g ∈ G(k) we can find a v such that g 6∈ Ev and such that G\Ev is affine. Thus the canonical section (Divisors, Definition 14.1) of OG(Ev) corresponds to a section sv of L which does not vanish at g and such

that Gsv is affine. This means that L is ample by definition (Properties, Definition 26.1). 0BF8 Lemma 8.8. Let k be a field. Let G be a locally algebraic group scheme over k. Then the center of G is a closed subgroup scheme of G. Proof. Let Aut(G) denote the contravariant functor on the category of schemes over k which associates to S/k the set of automorphisms of the base change GS as a group scheme over S. There is a natural transformation

G −→ Aut(G), g 7−→ inng sending an S-valued point g of G to the inner automorphism of G determined by g. The center C of G is by deﬁnition the kernel of this transformation, i.e., the functor which to S associates those g ∈ G(S) whose associated inner automorphism is trivial. The statement of the lemma is that this functor is representable by a closed subgroup scheme of G.

1Using the material in Divisors, Section 17 we could take as effective Cartier divisor E the norm of the effective Cartier divisor D along the finite locally free morphism 1 × π0 bypassing some of the arguments. GROUPOID SCHEMES 17

Choose an integer n ≥ 1. Let Gn ⊂ G be the nth infinitesimal neighbourhood of the identity element e of G. For every scheme S/k the base change Gn,S is the nth infinitesimal neighbourhood of eS : S → GS. Thus we see that there is a natural transformation Aut(G) → Aut(Gn) where the right hand side is the functor of automorphisms of Gn as a scheme (Gn isn’t in general a group scheme). Observe that Gn is the spectrum of an artinian local ring An with residue field k which has finite dimension as a k-vector space (Varieties, Lemma 20.2). Since every automorphism of Gn induces in particular an invertible linear map An → An, we obtain transformations of functors

G → Aut(G) → Aut(Gn) → GL(An) The ﬁnal group valued functor is representable, see Example 5.4, and the last arrow is visibly injective. Thus for every n we obtain a closed subgroup scheme

Hn = Ker(G → Aut(Gn)) = Ker(G → GL(An)). T As a ﬁrst approximation we set H = n≥1 Hn (scheme theoretic intersection). This is a closed subgroup scheme which contains the center C.

Let h be an S-valued point of H with S locally Noetherian. Then the automorphism innh induces the identity on all the closed subschemes Gn,S. Consider the kernel K = Ker(innh : GS → GS). This is a closed subgroup scheme of GS over S containing the closed subschemes Gn,S for n ≥ 1. This implies that K contains 0 an open neighbourhood of e(S) ⊂ GS, see Algebra, Remark 51.6. Let G ⊂ G be as in Proposition 7.11. Since G0 is geometrically irreducible, we conclude that K 0 0 0 contains GS (for any nonempty open U ⊂ Gk0 and any ﬁeld extension k /k we have −1 0 U · U = Gk0 , see proof of Lemma 7.9). Applying this with S = H we ﬁnd that G0 and H are subgroup schemes of G whose points commute: for any scheme S and any S-valued points g ∈ G0(S), h ∈ H(S) we have gh = hg in G(S).

Assume that k is algebraically closed. Then we can pick a k-valued point gi in each irreducible component Gi of G. Observe that in this case the connected components 0 of G are the irreducible components of G are the translates of G by our gi. We claim that \ C = H ∩ Ker(inng : G → G)(scheme theoretic intersection) i i Namely, C is contained in the right hand side. On the other hand, every S-valued 0 point h of the right hand side commutes with G and with gi hence with everything S 0 in G = G gi. The case of a general base ﬁeld k follows from the result for the algebraic closure k by descent. Namely, let A ⊂ Gk the closed subgroup scheme representing the center of Gk. Then we have

A ×Spec(k) Spec(k) = Spec(k) ×Spec(k) A as closed subschemes of G by the functorial nature of the center. Hence we see k⊗kk that A descends to a closed subgroup scheme Z ⊂ G by Descent, Lemma 34.2 (and Descent, Lemma 20.19). Then Z represents C (small argument omitted) and the proof is complete. GROUPOID SCHEMES 18

9. Abelian varieties 0BF9 An excellent reference for this material is Mumford’s book on abelian varieties, see [Mum70]. We encourage the reader to look there. There are many equivalent definitions; here is one. 03RODefinition 9.1. Let k be a field. An abelian variety is a group scheme over k which is also a proper, geometrically integral variety over k. We prove a few lemmas about this notion and then we collect all the results together in Proposition 9.11. 0BFA Lemma 9.2. Let k be a field. Let A be an abelian variety over k. Then A is projective.

Proof. This follows from Lemma 8.7 and More on Morphisms, Lemma 46.1. 0BFB Lemma 9.3. Let k be a field. Let A be an abelian variety over k. For any field extension K/k the base change AK is an abelian variety over K. Proof. Omitted. Note that this is why we insisted on A being geometrically integral; without that condition this lemma (and many others below) would be wrong. 0BFC Lemma 9.4. Let k be a field. Let A be an abelian variety over k. Then A is smooth over k. Proof. If k is perfect then this follows from Lemma 8.2 (characteristic zero) and Lemma 8.4 (positive characteristic). We can reduce the general case to this case by descent for smoothness (Descent, Lemma 20.27) and going to the perfect closure using Lemma 9.3. 0BFD Lemma 9.5. An abelian variety is an abelian group scheme, i.e., the group law is commutative. Proof. Let k be a field. Let A be an abelian variety over k. By Lemma 9.3 we may replace k by its algebraic closure. Consider the morphism −1 −1 h : A ×k A −→ A ×k A, (x, y) 7−→ (x, xyx y ) This is a morphism over A via the first projection on either side. Let e ∈ A(k) be the unit. Then we see that h|e×A is constant with value (e, e). By More on Morphisms, Lemma 40.3 there exists an open neighbourhood U ⊂ A of e such that h|U×A factors through some Z ⊂ U ×A finite over U. This means that for x ∈ U(k) the morphism A → A, y 7→ xyx−1y−1 takes finitely many values. Of course this means it is constant with value e. Thus (x, y) 7→ xyx−1y−1 is constant with value e on U × A which implies that the group law on A is abelian. 0BFE Lemma 9.6. Let k be a field. Let A be an abelian variety over k. Let L be an invertible OA-module. Then there is an isomorphism ∗ ∗ ∗ ∗ ∼ ∗ ∗ ∗ m1,2,3L ⊗ m1L ⊗ m2L ⊗ m3L = m1,2L ⊗ m1,3L ⊗ m2,3L

of invertible modules on A ×k A ×k A where mi ,...,i : A ×k A ×k A → A is the P 1 t morphism (x1, x2, x3) 7→ xij . GROUPOID SCHEMES 19

Proof. Apply the theorem of the cube (More on Morphisms, Theorem 30.8) to the diﬀerence ∗ ∗ ∗ ∗ ∗ ⊗−1 ∗ ⊗−1 ∗ ⊗−1 M = m1,2,3L ⊗ m1L ⊗ m2L ⊗ m3L ⊗ m1,2L ⊗ m1,3L ⊗ m2,3L This works because the restriction of M to A × A × e = A × A is equal to ∗ ∗ ∗ ∗ ⊗−1 ∗ ⊗−1 ∗ ⊗−1 ∼ n1,2L ⊗ n1L ⊗ n2L ⊗ n1,2L ⊗ n1L ⊗ n2L = OA×kA P where ni1,...,it : A ×k A → A is the morphism (x1, x2) 7→ xij . Similarly for A × e × A and e × A × A. 0BFF Lemma 9.7. Let k be a ﬁeld. Let A be an abelian variety over k. Let L be an invertible OA-module. Then [n]∗L =∼ L⊗n(n+1)/2 ⊗ ([−1]∗L)⊗n(n−1)/2 where [n]: A → A sends x to x + x + ... + x with n summands and where [−1] : A → A is the inverse of A.

Proof. Consider the morphism A → A ×k A ×k A, x 7→ (x, x, −x) where −x = [−1](x). Pulling back the relation of Lemma 9.6 we obtain L ⊗ L ⊗ L ⊗ [−1]∗L =∼ [2]∗L which proves the result for n = 2. By induction assume the result holds for 1, 2, . . . , n. Then consider the morphism A → A ×k A ×k A, x 7→ (x, x, [n − 1]x). Pulling back the relation of Lemma 9.6 we obtain [n + 1]∗L ⊗ L ⊗ L ⊗ [n − 1]∗L =∼ [2]∗L ⊗ [n]∗L ⊗ [n]∗L

and the result follows by elementary arithmetic. 0BFG Lemma 9.8. Let k be a field. Let A be an abelian variety over k. Let [d]: A → A be the multiplication by d. Then [d] is finite locally free of degree d2 dim(A). Proof. By Lemma 9.2 (and More on Morphisms, Lemma 46.1) we see that A has an ample invertible module L. Since [−1] : A → A is an automorphism, we see ∗ ∗ that [−1] L is an ample invertible OX -module as well. Thus N = L ⊗ [−1] L is 2 ample, see Properties, Lemma 26.5. Since N =∼ [−1]∗N we see that [d]∗N =∼ N ⊗d by Lemma 9.7. To get a contradiction C ⊂ X be a proper curve contained in a fibre of [d]. Then ⊗d2 ∼ N |C = OC is an ample invertible OC -module of degree 0 which contradicts Varieties, Lemma 43.14 for example. (You can also use Varieties, Lemma 44.9.) Thus every fibre of [d] has dimension 0 and hence [d] is finite for example by Cohomology of Schemes, Lemma 21.1. Moreover, since A is smooth over k by Lemma 9.4 we see that [d]: A → A is flat by Algebra, Lemma 128.1 (we also use that schemes smooth over fields are regular and that regular rings are Cohen- Macaulay, see Varieties, Lemma 25.3 and Algebra, Lemma 106.3). Thus [d] is finite flat hence finite locally free by Morphisms, Lemma 48.2. Finally, we come to the formula for the degree. By Varieties, Lemma 44.11 we see that

degN ⊗d2 (A) = deg([d]) degN (A) GROUPOID SCHEMES 20

2 Since the degree of A with respect to N ⊗d , respectively N is the coeﬃcient of ndim(A) in the polynomial

2 n 7−→ χ(A, N ⊗nd ), respectively n 7−→ χ(A, N ⊗n)

2 dim(A) we see that deg([d]) = d . 0BFH Lemma 9.9. Let k be a field. Let A be a nonzero abelian variety over k. Then [d]: A → A is étale if and only if d is invertible in k. Proof. Observe that [d](x + y) = [d](x) + [d](y). Since translation by a point is an automorphism of A, we see that the set of points where [d]: A → A is étale is either empty or equal to A (some details omitted). Thus it suffices to check whether [d] is étale at the unit e ∈ A(k). Since we know that [d] is finite locally free (Lemma 9.8) to see that it is étale at e is equivalent to proving that d[d]: TA/k,e → TA/k,e is injective. See Varieties, Lemma 16.8 and Morphisms, Lemma 36.16. By Lemma 6.4 we see that d[d] is given by multiplication by d on TA/k,e. 0C0Y Lemma 9.10. Let k be a field of characteristic p > 0. Let A be an abelian variety over k. The fibre of [p]: A → A over 0 has at most pg distinct points. Proof. To prove this, we may and do replace k by the algebraic closure. By Lemma 6.4 the derivative of [p] is multiplication by p as a map TA/k,e → TA/k,e and hence is zero (compare with proof of Lemma 9.9). Since [p] commutes with translation we conclude that the derivative of [p] is everywhere zero, i.e., that the induced map ∗ [p] ΩA/k → ΩA/k is zero. Looking at generic points, we find that the corresponding ∗ map [p] : k(A) → k(A) of function fields induces the zero map on Ωk(A)/k. Let t1, . . . , tg be a p-basis of k(A) over k (More on Algebra, Definition 46.1 and Lemma ∗ 46.2). Then [p] (ti) has a pth root by Algebra, Lemma 158.2. We conclude that p p ∗ k(A)[x1, . . . , xg]/(x1 −t1, . . . , xg −tg) is a subextension of [p] : k(A) → k(A). Thus −1 we can find an affine open U ⊂ A such that ti ∈ OA(U) and xi ∈ OA([p] (U)). We obtain a factorization

−1 π1 p p π2 [p] (U) −→ Spec(O(U)[x1, . . . , xg]/(x1 − t1, . . . , xg − tg)) −→ U

of [p] over U. After shrinking U we may assume that π1 is finite locally free (for example by generic flatness – actually it is already finite locally free in our case). By 2g g Lemma 9.8 we see that [p] has degree p . Since π2 has degree p we see that π1 has g degree p as well. The morphism π2 is a universal homeomorphism hence the fibres are singletons. We conclude that the (set theoretic) fibres of [p]−1(U) → U are the g fibres of π1. Hence they have at most p elements. Since [p] is a homomorphism of group schemes over k, the fibre of [p]: A(k) → A(k) has the same cardinality for every a ∈ A(k) and the proof is complete. 03RP Proposition 9.11. Let A be an abelian variety over a field k. Then Wonderfully (1) A is projective over k, explained in (2) A is a commutative group scheme, [Mum70]. (3) the morphism [n]: A → A is surjective for all n ≥ 1, (4) if k is algebraically closed, then A(k) is a divisible abelian group, (5) A[n] = Ker([n]: A → A) is a finite group scheme of degree n2 dim A over k, (6) A[n] is étale over k if and only if n ∈ k∗, (7) if n ∈ k∗ and k is algebraically closed, then A(k)[n] =∼ (Z/nZ)⊕2 dim(A), GROUPOID SCHEMES 21

(8) if k is algebraically closed of characteristic p > 0, then there exists an integer 0 ≤ f ≤ dim(A) such that A(k)[pm] =∼ (Z/pmZ)⊕f for all m ≥ 1.

Proof. Part (1) follows from Lemma 9.2. Part (2) follows from Lemma 9.5. Part (3) follows from Lemma 9.8. If k is algebraically closed then surjective morphisms of varieties over k induce surjective maps on k-rational points, hence (4) follows from (3). Part (5) follows from Lemma 9.8 and the fact that a base change of a ﬁnite locally free morphism of degree N is a ﬁnite locally free morphism of degree N. Part (6) follows from Lemma 9.9. Namely, if n is invertible in k, then [n] is étale and hence A[n] is étale over k. On the other hand, if n is not invertible in k, then [n] is not étale at e and it follows that A[n] is not étale over k at e (use Morphisms, Lemmas 36.16 and 35.15).

Assume k is algebraically closed. Set g = dim(A). Proof of (7). Let ` be a prime number which is invertible in k. Then we see that

A[`](k) = A(k)[`]

is a ﬁnite abelian group, annihilated by `, of order `2g. It follows that it is isomorphic to (Z/`Z)2g by the structure theory for ﬁnite abelian groups. Next, we consider the short exact sequence

0 → A(k)[`] → A(k)[`2] −→` A(k)[`] → 0

Arguing similarly as above we conclude that A(k)[`2] =∼ (Z/`2Z)2g. By induction on the exponent we ﬁnd that A(k)[`m] =∼ (Z/`mZ)2g. For composite integers n prime to the characteristic of k we take primary parts and we ﬁnd the correct shape of the n-torsion in A(k). The proof of (8) proceeds in exactly the same way, using ∼ ⊕f that Lemma 9.10 gives A(k)[p] = (Z/pZ) for some 0 ≤ f ≤ g.

10. Actions of group schemes 022Y Let (G, m) be a group and let V be a set. Recall that a (left) action of G on V is given by a map a : G × V → V such that (1) (associativity) a(m(g, g0), v) = a(g, a(g0, v)) for all g, g0 ∈ G and v ∈ V , and (2) (identity) a(e, v) = v for all v ∈ V . We also say that V is a G-set (this usually means we drop the a from the notation – which is abuse of notation). A map of G-sets ψ : V → V 0 is any set map such that ψ(a(g, v)) = a(g, ψ(v)) for all v ∈ V .

022ZDeﬁnition 10.1. Let S be a scheme. Let (G, m) be a group scheme over S.

(1) An action of G on the scheme X/S is a morphism a : G ×S X → X over S such that for every T/S the map a : G(T ) × X(T ) → X(T ) deﬁnes the structure of a G(T )-set on X(T ). (2) Suppose that X, Y are schemes over S each endowed with an action of G. An equivariant or more precisely a G-equivariant morphism ψ : X → Y is a morphism of schemes over S such that for every T/S the map ψ : X(T ) → Y (T ) is a morphism of G(T )-sets. GROUPOID SCHEMES 22

In situation (1) this means that the diagrams

G ×S G ×S X / G ×S X G ×S X / X 1G×a O a ;

(10.1.1)03LD m×1X a e×1X 1X a G ×S X / X X are commutative. In situation (2) this just means that the diagram

G ×S X / G ×S Y id×ψ a a ψ X / Y commutes. 07S1Definition 10.2. Let S, G → S, and X → S as in Definition 10.1. Let a : G ×S X → X be an action of G on X/S. We say the action is free if for every scheme T over S the action a : G(T ) × X(T ) → X(T ) is a free action of the group G(T ) on the set X(T ). 07S2 Lemma 10.3. Situation as in Definition 10.2, The action a is free if and only if

G ×S X → X ×S X, (g, x) 7→ (a(g, x), x) is a monomorphism.

Proof. Immediate from the definitions. 11. Principal homogeneous spaces 0497 In Cohomology on Sites, Definition 4.1 we have defined a torsor for a sheaf of groups on a site. Suppose τ ∈ {Zariski, etale,´ smooth, syntomic, fppf} is a topology and (G, m) is a group scheme over S. Since τ is stronger than the canonical topology (see Descent, Lemma 10.7) we see that G (see Sites, Definition 12.3) is a sheaf of groups on (Sch/S)τ . Hence we already know what it means to have a torsor for G on (Sch/S)τ . A special situation arises if this sheaf is representable. In the following definitions we define directly what it means for the representing scheme to be a G-torsor. 0498Definition 11.1. Let S be a scheme. Let (G, m) be a group scheme over S. Let X be a scheme over S, and let a : G ×S X → X be an action of G on X. (1) We say X is a pseudo G-torsor or that X is formally principally homogeneous under G if the induced morphism of schemes G ×S X → X ×S X, (g, x) 7→ (a(g, x), x) is an isomorphism of schemes over S. (2) A pseudo G-torsor X is called trivial if there exists an G-equivariant isomorphism G → X over S where G acts on G by left multiplication.

0 It is clear that if S → S is a morphism of schemes then the pullback XS0 of a 0 pseudo G-torsor over S is a pseudo GS0 -torsor over S . 0499 Lemma 11.2. In the situation of Deﬁnition 11.1. (1) The scheme X is a pseudo G-torsor if and only if for every scheme T over S the set X(T ) is either empty or the action of the group G(T ) on X(T ) is simply transitive. GROUPOID SCHEMES 23

(2) A pseudo G-torsor X is trivial if and only if the morphism X → S has a section.

Proof. Omitted. 049ADeﬁnition 11.3. Let S be a scheme. Let (G, m) be a group scheme over S. Let X be a pseudo G-torsor over S. (1) We say X is a principal homogeneous space or a G-torsor if there exists a 2 fpqc covering {Si → S}i∈I such that each XSi → Si has a section (i.e., is

a trivial pseudo GSi -torsor). (2) Let τ ∈ {Zariski, etale,´ smooth, syntomic, fppf}. We say X is a G-torsor in the τ topology, or a τ G-torsor, or simply a τ torsor if there exists a τ

covering {Si → S}i∈I such that each XSi → Si has a section. (3) If X is a G-torsor, then we say that it is quasi-isotrivial if it is a torsor for the étale topology. (4) If X is a G-torsor, then we say that it is locally trivial if it is a torsor for the Zariski topology. We sometimes say “let X be a G-torsor over S” to indicate that X is a scheme over S equipped with an action of G which turns it into a principal homogeneous space over S. Next we show that this agrees with the notation introduced earlier when both apply. 049B Lemma 11.4. Let S be a scheme. Let (G, m) be a group scheme over S. Let X be a scheme over S, and let a : G ×S X → X be an action of G on X. Let τ ∈ {Zariski, etale,´ smooth, syntomic, fppf}. Then X is a G-torsor in the τ- topology if and only if X is a G-torsor on (Sch/S)τ .

Proof. Omitted. 049CRemark 11.5. Let (G, m) be a group scheme over the scheme S. In this situation we have the following natural types of questions: (1) If X → S is a pseudo G-torsor and X → S is surjective, then is X necessarily a G-torsor? (2) Is every G-torsor on (Sch/S)fppf representable? In other words, does every G-torsor come from a fppf G-torsor? (3) Is every G-torsor an fppf (resp. smooth, resp. étale, resp. Zariski) torsor? In general the answers to these questions is no. To get a positive answer we need to impose additional conditions on G → S. For example: If S is the spectrum of a ﬁeld, then the answer to (1) is yes because then {X → S} is a fpqc covering trivializing X. If G → S is aﬃne, then the answer to (2) is yes (insert future reference here). If G = GLn,S then the answer to (3) is yes and in fact any GLn,S-torsor is locally trivial (insert future reference here).

12. Equivariant quasi-coherent sheaves 03LE We think of “functions” as dual to “space”. Thus for a morphism of spaces the map on functions goes the other way. Moreover, we think of the sections of a sheaf

2This means that the default type of torsor is a pseudo torsor which is trivial on an fpqc covering. This is the deﬁnition in [ABD+66, Exposé IV, 6.5]. It is a little bit inconvenient for us as we most often work in the fppf topology. GROUPOID SCHEMES 24

of modules as “functions”. This leads us naturally to the direction of the arrows chosen in the following deﬁnition. 03LFDeﬁnition 12.1. Let S be a scheme, let (G, m) be a group scheme over S, and let a : G ×S X → X be an action of the group scheme G on X/S.A G-equivariant quasi-coherent OX -module, or simply an equivariant quasi-coherent OX -module, is

a pair (F, α), where F is a quasi-coherent OX -module, and α is a OG×S X -module map ∗ ∗ α : a F −→ pr1F

where pr1 : G ×S X → X is the projection such that (1) the diagram

∗ ∗ ∗ (1G × a) pr F / pr F 1 pr∗ α 2 O 12 O ∗ ∗ (1G×a) α (m×1X ) α

∗ ∗ ∗ ∗ (1G × a) a F (m × 1X ) a F

is a commutative in the category of OG×S G×S X -modules, and (2) the pullback ∗ (e × 1X ) α : F −→ F is the identity map. For explanation compare with the relevant diagrams of Equation (10.1.1).

∗ Note that the commutativity of the ﬁrst diagram guarantees that (e × 1X ) α is an idempotent operator on F, and hence condition (2) is just the condition that it is an isomorphism. 03LG Lemma 12.2. Let S be a scheme. Let G be a group scheme over S. Let f : Y → X be a G-equivariant morphism between S-schemes endowed with G-actions. Then ∗ ∗ ∗ pullback f given by (F, α) 7→ (f F, (1G ×f) α) deﬁnes a functor from the category of G-equivariant quasi-coherent OX -modules to the category of G-equivariant quasi- coherent OY -modules.

Proof. Omitted. Let us give an example. 0EKJExample 12.3. Let A be a Z-graded ring, i.e., A comes with a direct sum decom- L position A = n∈Z An and An · Am ⊂ An+m. Set X = Spec(A). Then we obtain a Gm-action a : Gm × X −→ X by the ring map µ : A → A ⊗ Z[x, x−1], f 7→ f ⊗ xdeg(f). Namely, to check this we have to verify that

−1 A µ / A ⊗ Z[x, x ]

µ µ⊗1

1⊗m A ⊗ Z[x, x−1] / A ⊗ Z[x, x−1] ⊗ Z[x, x−1] where m(x) = x⊗x, see Example 5.1. This is immediately clear when evaluating on a homogeneous element. Suppose that M is a graded A-module. Then we obtain GROUPOID SCHEMES 25

a Gm-equivariant quasi-coherent OX -module F = Mf by using α as in Deﬁnition 12.1 corresponding to the A ⊗ Z[x, x−1]-module map −1 −1 M ⊗A,µ (A ⊗ Z[x, x ]) −→ M ⊗A,idA⊗1 (A ⊗ Z[x, x ]) sending m ⊗ 1 ⊗ 1 to m ⊗ 1 ⊗ xdeg(m) for m ∈ M homogeneous.

0EKK Lemma 12.4. Let a : Gm × X → X be an action on an aﬃne scheme. Then X is the spectrum of a Z-graded ring and the action is as in Example 12.3.

Proof. Let f ∈ A = Γ(X, OX ). Then we can write ] X n −1 a (f) = fn ⊗ x in A ⊗ Z[x, x ] = Γ(Gm × X, OG ×X ) n∈Z m

as a ﬁnite sum with fn in A uniquely determined. Thus we obtain maps A → A, P f 7→ fn. Since a is an action, if we evaluate at x = 1, we see f = fn. Since a is an action we ﬁnd that X m n X n n (fn)m ⊗ x ⊗ x = fnx ⊗ x

(compare with computation in Example 12.3). Thus (fn)m = 0 if n 6= m and (fn)n = fn. Thus if we set

An = {f ∈ A | fn = f} P then we get A = An. On the other hand, the sum has to be direct since f = 0 implies fn = 0 in the situation above.

0EKL Lemma 12.5. Let A be a graded ring. Let X = Spec(A) with action a : Gm ×X → X as in Example 12.3. Let F be a Gm-equivariant quasi-coherent OX -module. Then M = Γ(X, F) has a canonical grading such that it is a graded A-module and such that the isomorphism Mf → F (Schemes, Lemma 7.4) is an isomorphism of Gm-equivariant modules where the Gm-equivariant structure on Mf is the one from Example 12.3. Proof. You can either prove this by repeating the arguments of Lemma 12.4 for the 0 module M. Alternatively, you can consider the scheme (X , OX0 ) = (X, OX ⊕ F) 0 where F is viewed as an ideal of square zero. There is a natural action a : Gm × X0 → X0 deﬁned using the action on X and on F. Then apply Lemma 12.4 to X0 and conclude. (The nice thing about this argument is that it immediately shows that the grading on A and M are compatible, i.e., that M is a graded A-module.) Details omitted. 13. Groupoids 0230 Recall that a groupoid is a category in which every morphism is an isomorphism, see Categories, Deﬁnition 2.5. Hence a groupoid has a set of objects Ob, a set of arrows Arrows, a source and target map s, t : Arrows → Ob, and a composition law c : Arrows ×s,Ob,t Arrows → Arrows. These maps satisfy exactly the following axioms

(1) (associativity) c ◦ (1, c) = c ◦ (c, 1) as maps Arrows ×s,Ob,t Arrows ×s,Ob,t Arrows → Arrows, (2) (identity) there exists a map e : Ob → Arrows such that (a) s ◦ e = t ◦ e = id as maps Ob → Ob, (b) c ◦ (1, e ◦ s) = c ◦ (e ◦ t, 1) = 1 as maps Arrows → Arrows, GROUPOID SCHEMES 26

(3) (inverse) there exists a map i : Arrows → Arrows such that (a) s ◦ i = t, t ◦ i = s as maps Arrows → Ob, and (b) c ◦ (1, i) = e ◦ t and c ◦ (i, 1) = e ◦ s as maps Arrows → Arrows. If this is the case the maps e and i are uniquely determined and i is a bijection. Note that if (Ob0, Arrows0, s0, t0, c0) is a second groupoid category, then a functor f :(Ob, Arrows, s, t, c) → (Ob0, Arrows0, s0, t0, c0) is given by a pair of set maps f : Ob → Ob0 and f : Arrows → Arrows0 such that s0 ◦ f = f ◦ s, t0 ◦ f = f ◦ t, and c0 ◦ (f, f) = f ◦ c. The compatibility with identity and inverse is automatic. We will use this below. (Warning: The compatibility with identity has to be imposed in the case of general categories.) 0231Definition 13.1. Let S be a scheme. (1) A groupoid scheme over S, or simply a groupoid over S is a quintuple (U, R, s, t, c) where U and R are schemes over S, and s, t : R → U and c : R ×s,U,t R → R are morphisms of schemes over S with the following property: For any scheme T over S the quintuple (U(T ),R(T ), s, t, c) is a groupoid category in the sense described above. (2) A morphism f :(U, R, s, t, c) → (U 0,R0, s0, t0, c0) of groupoid schemes over S is given by morphisms of schemes f : U → U 0 and f : R → R0 with the following property: For any scheme T over S the maps f define a functor from the groupoid category (U(T ),R(T ), s, t, c) to the groupoid category (U 0(T ),R0(T ), s0, t0, c0). Let (U, R, s, t, c) be a groupoid over S. Note that, by the remarks preceding the definition and the Yoneda lemma, there are unique morphisms of schemes e : U → R and i : R → R over S such that for every scheme T over S the induced map e : U(T ) → R(T ) is the identity, and i : R(T ) → R(T ) is the inverse of the groupoid category. The septuple (U, R, s, t, c, e, i) satisfies commutative diagrams corresponding to each of the axioms (1), (2)(a), (2)(b), (3)(a) and (3)(b) above, and conversely given a septuple with this property the quintuple (U, R, s, t, c) is a groupoid scheme. Note that i is an isomorphism, and e is a section of both s and t. Moreover, given a groupoid scheme over S we denote

j = (t, s): R −→ U ×S U which is compatible with our conventions in Section 3 above. We sometimes say “let (U, R, s, t, c, e, i) be a groupoid over S” to stress the existence of identity and inverse. 0232 Lemma 13.2. Given a groupoid scheme (U, R, s, t, c) over S the morphism j : R → U ×S U is a pre-equivalence relation.

Proof. Omitted. This is a nice exercise in the deﬁnitions.

0233 Lemma 13.3. Given an equivalence relation j : R → U ×S U over S there is a unique way to extend it to a groupoid (U, R, s, t, c) over S.

Proof. Omitted. This is a nice exercise in the deﬁnitions. GROUPOID SCHEMES 27

02YE Lemma 13.4. Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. In the commutative diagram

U : d t t

R o R ×s,U,t R / R pr0 c

s pr1 s

t s URo / U the two lower squares are ﬁbre product squares. Moreover, the triangle on top (which is really a square) is also cartesian.

Proof. Omitted. Exercise in the deﬁnitions and the functorial point of view in algebraic geometry.

03C6 Lemma 13.5. Let S be a scheme. Let (U, R, s, t, c, e, i) be a groupoid over S. The diagram

pr1 (13.5.1)03C7 R × R / R t U t,U,t / / pr0

(pr0,c◦(i,1)) idR idU c R × R / R t U s,U,t / / pr0 pr1 s s R / U t / is commutative. The two top rows are isomorphic via the vertical maps given. The two lower left squares are cartesian.

Proof. The commutativity of the diagram follows from the axioms of a groupoid. Note that, in terms of groupoids, the top left vertical arrow assigns to a pair of morphisms (α, β) with the same target, the pair of morphisms (α, α−1 ◦ β). In any groupoid this deﬁnes a bijection between Arrows ×t,Ob,t Arrows and Arrows ×s,Ob,t Arrows. Hence the second assertion of the lemma. The last assertion follows from Lemma 13.4.

0DT8 Lemma 13.6. Let (U, R, s, t, c) be a groupoid over a scheme S. Let S0 → S be a 0 0 0 0 morphism. Then the base changes U = S ×S U, R = S ×S R endowed with the base changes s0, t0, c0 of the morphisms s, t, c form a groupoid scheme (U 0,R0, s0, t0, c0) over S0 and the projections determine a morphism (U 0,R0, s0, t0, c0) → (U, R, s, t, c) of groupoid schemes over S.

0 0 0 Proof. Omitted. Hint: R ×s0,U 0,t0 R = S ×S (R ×s,U,t R).

14. Quasi-coherent sheaves on groupoids 03LH See the introduction of Section 12 for our choices in direction of arrows. GROUPOID SCHEMES 28

03LIDeﬁnition 14.1. Let S be a scheme, let (U, R, s, t, c) be a groupoid scheme over S.A quasi-coherent module on (U, R, s, t, c) is a pair (F, α), where F is a quasi- coherent OU -module, and α is a OR-module map α : t∗F −→ s∗F such that (1) the diagram ∗ ∗ ∗ ∗ pr1t F ∗ / pr1s F pr1 α

∗ ∗ ∗ ∗ pr0s F c s F e :

∗ ∗ pr0 α c α ∗ ∗ ∗ ∗ pr0t F c t F

is a commutative in the category of OR×s,U,tR-modules, and (2) the pullback e∗α : F −→ F is the identity map. Compare with the commutative diagrams of Lemma 13.4. The commutativity of the first diagram forces the operator e∗α to be idempotent. Hence the second condition can be reformulated as saying that e∗α is an isomorphism. In fact, the condition implies that α is an isomorphism. 077Q Lemma 14.2. Let S be a scheme, let (U, R, s, t, c) be a groupoid scheme over S. If (F, α) is a quasi-coherent module on (U, R, s, t, c) then α is an isomorphism. Proof. Pull back the commutative diagram of Definition 14.1 by the morphism ∗ ∗ ∗ (i, 1) : R → R ×s,U,t R. Then we see that i α ◦ α = s e α. Pulling back by the morphism (1, i) we obtain the relation α ◦ i∗α = t∗e∗α. By the second assumption ∗ these morphisms are the identity. Hence i α is an inverse of α. 03LJ Lemma 14.3. Let S be a scheme. Consider a morphism f :(U, R, s, t, c) → (U 0,R0, s0, t0, c0) of groupoid schemes over S. Then pullback f ∗ given by (F, α) 7→ (f ∗F, f ∗α) defines a functor from the category of quasi-coherent sheaves on (U 0,R0, s0, t0, c0) to the category of quasi-coherent sheaves on (U, R, s, t, c).

Proof. Omitted. 09VH Lemma 14.4. Let S be a scheme. Consider a morphism f :(U, R, s, t, c) → (U 0,R0, s0, t0, c0) of groupoid schemes over S. Assume that (1) f : U → U 0 is quasi-compact and quasi-separated, (2) the square R / R0 f t t0 f U / U 0 is cartesian, and GROUPOID SCHEMES 29

(3) s0 and t0 are ﬂat.

Then pushforward f∗ given by

(F, α) 7→ (f∗F, f∗α) deﬁnes a functor from the category of quasi-coherent sheaves on (U, R, s, t, c) to the category of quasi-coherent sheaves on (U 0,R0, s0, t0, c0) which is right adjoint to pullback as deﬁned in Lemma 14.3.

0 Proof. Since U → U is quasi-compact and quasi-separated we see that f∗ trans- forms quasi-coherent sheaves into quasi-coherent sheaves (Schemes, Lemma 24.1). Moreover, since the squares

R / R0 R / R0 f f t t0 and s s0 f f U / U 0 U / U 0 0 ∗ ∗ 0 ∗ ∗ are cartesian we ﬁnd that (t ) f∗F = f∗t F and (s ) f∗F = f∗s F , see Coho- mology of Schemes, Lemma 5.2. Thus it makes sense to think of f∗α as a map 0 ∗ 0 ∗ (t ) f∗F → (s ) f∗F. A similar argument shows that f∗α satisﬁes the cocycle condition. The functor is adjoint to the pullback functor since pullback and pushforward on modules on ringed spaces are adjoint. Some details omitted. 077R Lemma 14.5. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. The category of quasi-coherent modules on (U, R, s, t, c) has colimits.

Proof. Let i 7→ (Fi, αi) be a diagram over the index category I. We can form the colimit F = colim Fi which is a quasi-coherent sheaf on U, see Schemes, Section ∗ ∗ 24. Since colimits commute with pullback we see that s F = colim s Fi and sim- ∗ ∗ ilarly t F = colim t Fi. Hence we can set α = colim αi. We omit the proof that (F, α) is the colimit of the diagram in the category of quasi-coherent modules on (U, R, s, t, c). 077S Lemma 14.6. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. If s, t are ﬂat, then the category of quasi-coherent modules on (U, R, s, t, c) is abelian. Proof. Let ϕ :(F, α) → (G, β) be a homomorphism of quasi-coherent modules on (U, R, s, t, c). Since s is ﬂat we see that 0 → s∗ Ker(ϕ) → s∗F → s∗G → s∗ Coker(ϕ) → 0 is exact and similarly for pullback by t. Hence α and β induce isomorphisms κ : t∗ Ker(ϕ) → s∗ Ker(ϕ) and λ : t∗ Coker(ϕ) → s∗ Coker(ϕ) which satisfy the cocycle condition. Then it is straightforward to verify that (Ker(ϕ), κ) and (Coker(ϕ), λ) are a kernel and cokernel in the category of quasi-coherent modules on (U, R, s, t, c). Moreover, the condition Coim(ϕ) = Im(ϕ) follows because it holds over U.

15. Colimits of quasi-coherent modules 07TS In this section we prove some technical results saying that under suitable assumptions every quasi-coherent module on a groupoid is a ﬁltered colimit of “small” quasi-coherent modules. GROUPOID SCHEMES 30

07TR Lemma 15.1. Let (U, R, s, t, c) be a groupoid scheme over S. Assume s, t are ﬂat, quasi-compact, and quasi-separated. For any quasi-coherent module G on U, there ∗ ∗ ∗ ∗ ∗ exists a canonical isomorphism α : t t∗s G → s t∗s G which turns (t∗s G, α) into a quasi-coherent module on (U, R, s, t, c). This construction deﬁnes a functor

QCoh(OU ) −→ QCoh(U, R, s, t, c) which is a right adjoint to the forgetful functor (F, β) 7→ F.

Proof. The pushforward of a quasi-coherent module along a quasi-compact and quasi-separated morphism is quasi-coherent, see Schemes, Lemma 24.1. Hence ∗ t∗s G is quasi-coherent. With notation as in Lemma 13.4 we have

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ t t∗s G = pr0,∗c s G = pr0,∗pr1s G = s t∗s G

The middle equality because s ◦ c = s ◦ pr1 as morphisms R ×s,U,t R → U, and the ﬁrst and the last equality because we know that base change and pushforward commute in these steps by Cohomology of Schemes, Lemma 5.2. To verify the cocycle condition of Deﬁnition 14.1 for α and the adjointness property we describe the construction G 7→ (G, α) in another way. Consider the groupoid scheme (R,R×s,U,sR, pr0, pr1, pr02) associated to the equivalence relation R×s,U,sR on R, see Lemma 13.3. There is a morphism

f :(R,R ×s,U,s R, pr1, pr0, pr02) −→ (U, R, s, t, c)

of groupoid schemes given by t : R → U and R ×t,U,t R → R given by (r0, r1) 7→ −1 r0 ◦ r1 (we omit the veriﬁcation of the commutativity of the required diagrams). Since t, s : R → U are quasi-compact, quasi-separated, and ﬂat, and since we have a cartesian square R × R R s,U,s −1/ (r0,r1)7→r0◦r1 pr0 t t R / U by Lemma 13.5 it follows that Lemma 14.4 applies to f. Note that

QCoh(R,R ×s,U,s R, pr1, pr0, pr02) = QCoh(OU ) by the theory of descent of quasi-coherent sheaves as {t : R → U} is an fpqc covering, see Descent, Proposition 5.2. Observe that pullback along f agrees with the forgetful functor and that pushforward agrees with the construction that assigns to G the pair (G, α). We omit the precise veriﬁcations. Thus the lemma follows from Lemma 14.4.

07TT Lemma 15.2. Let f : Y → X be a morphism of schemes. Let F be a quasi- ∗ coherent OX -module, let G be a quasi-coherent OY -module, and let ϕ : G → f F be a module map. Assume (1) ϕ is injective, (2) f is quasi-compact, quasi-separated, ﬂat, and surjective, (3) X, Y are locally Noetherian, and (4) G is a coherent OY -module. GROUPOID SCHEMES 31

Then F ∩ f∗G deﬁned as the pullback ∗ F / f∗f F O O

F ∩ f∗G / f∗G

is a coherent OX -module. Proof. We will freely use the characterization of coherent modules of Cohomology of Schemes, Lemma 9.1 as well as the fact that coherent modules form a Serre subcategory of QCoh(OX ), see Cohomology of Schemes, Lemma 9.3. If f has a ∗ ∗ ∗ section σ, then we see that F ∩f∗G is contained in the image of σ G → σ f F = F, hence coherent. In general, to show that F ∩ f∗G is coherent, it suffices the show ∗ that f (F ∩ f∗G) is coherent (see Descent, Lemma 7.1). Since f is flat this is ∗ ∗ equal to f F ∩ f f∗G. Since f is flat, quasi-compact, and quasi-separated we see ∗ ∗ f f∗G = p∗q G where p, q : Y ×X Y → Y are the projections, see Cohomology of Schemes, Lemma 5.2. Since p has a section we win. Let S be a scheme. Let (U, R, s, t, c) be a groupoid in schemes over S. Assume that U is locally Noetherian. In the lemma below we say that a quasi-coherent sheaf (F, α) on (U, R, s, t, c) is coherent if F is a coherent OU -module. 07TU Lemma 15.3. Let (U, R, s, t, c) be a groupoid scheme over S. Assume that (1) U, R are Noetherian, (2) s, t are flat, quasi-compact, and quasi-separated. Then every quasi-coherent module (F, α) on (U, R, s, t, c) is a filtered colimit of coherent modules. Proof. We will use the characterization of Cohomology of Schemes, Lemma 9.1 of coherent modules on locally Noetherian scheme without further mention. Write F = colim Hi with Hi coherent, see Properties, Lemma 22.7. Given a quasi- ∗ coherent sheaf H on U we denote t∗s H the quasi-coherent sheaf on (U, R, s, t, c) ∗ of Lemma 15.1. There is an adjunction map F → t∗s F in QCoh(U, R, s, t, c). Consider the pullback diagram ∗ F / t∗s F O O

∗ Fi / t∗s Hi ∗ in other words Fi = F ∩ t∗s Hi. Then Fi is coherent by Lemma 15.2. On the other hand, the diagram above is a pullback diagram in QCoh(U, R, s, t, c) also as restriction to U is an exact functor by (the proof of) Lemma 14.6. Finally, because t is quasi-compact and quasi-separated we see that t∗ commutes with colimits (see ∗ Cohomology of Schemes, Lemma 6.1). Hence t∗s F = colim t∗Hi and hence F = colim Fi as desired. Here is a curious lemma that is useful when working with groupoids on ﬁelds. In fact, this is the standard argument to prove that any representation of an algebraic group is a colimit of ﬁnite dimensional representations. 07TV Lemma 15.4. Let (U, R, s, t, c) be a groupoid scheme over S. Assume that GROUPOID SCHEMES 32

(1) U, R are affine, (2) there exist ei ∈ OR(R) such that every element g ∈ OR(R) can be uniquely P ∗ written as s (fi)ei for some fi ∈ OU (U). Then every quasi-coherent module (F, α) on (U, R, s, t, c) is a filtered colimit of finite type quasi-coherent modules.

Proof. The assumption means that OR(R) is a free OU (U)-module via s with basis ∗ L ei. Hence for any quasi-coherent OU -module G we see that s G(R) = i G(U)ei. We will write s(−) to indicate pullback of sections by s and similarly for other morphisms. Let (F, α) be a quasi-coherent module on (U, R, s, t, c). Let σ ∈ F(U). By the above we can write X α(t(σ)) = s(σi)ei

for some unique σi ∈ F(U) (all but ﬁnitely many are zero of course). We can also write X c(ei) = pr1(fij)pr0(ej)

as functions on R ×s,U,t R. Then the commutativity of the diagram in Deﬁnition 14.1 means that X X pr1(α(t(σi)))pr0(ei) = pr1(s(σi)fij)pr0(ej)

(calculation omitted). Picking off the coefficients of pr0(el) we see that α(t(σl)) = P s(σi)fil. Hence the submodule G ⊂ F generated by the elements σi defines a finite type quasi-coherent module preserved by α. Hence it is a subobject of F in QCoh(U, R, s, t, c). This submodule contains σ (as one sees by pulling back the first relation by e). Hence we win. We suggest the reader skip the rest of this section. Let S be a scheme. Let (U, R, s, t, c) be a groupoid in schemes over S. Let κ be a cardinal. In the following we will say that a quasi-coherent sheaf (F, α) on (U, R, s, t, c) is κ-generated if F is a κ-generated OU -module, see Properties, Definition 23.1. 077T Lemma 15.5. Let (U, R, s, t, c) be a groupoid scheme over S. Let κ be a cardinal. There exists a set T and a family (Ft, αt)t∈T of κ-generated quasi-coherent modules on (U, R, s, t, c) such that every κ-generated quasi-coherent module on (U, R, s, t, c) is isomorphic to one of the (Ft, αt). Proof. For each quasi-coherent module F on U there is a (possibly empty) set of maps α : t∗F → s∗F such that (F, α) is a quasi-coherent modules on (U, R, s, t, c). By Properties, Lemma 23.2 there exists a set of isomorphism classes of κ-generated quasi-coherent OU -modules. 077U Lemma 15.6. Let (U, R, s, t, c) be a groupoid scheme over S. Assume that s, t are flat. There exists a cardinal κ such that every quasi-coherent module (F, α) on (U, R, s, t, c) is the directed colimit of its κ-generated quasi-coherent submodules. Proof. In the statement of the lemma and in this proof a submodule of a quasi- coherent module (F, α) is a quasi-coherent submodule G ⊂ F such that α(t∗G) = s∗G as subsheaves of s∗F. This makes sense because since s, t are flat the pullbacks s∗ and t∗ are exact, i.e., preserve subsheaves. The proof will be a repeat of the proof of Properties, Lemma 23.3. We urge the reader to read that proof first. GROUPOID SCHEMES 33

S Choose an aﬃne open covering U = i∈I Ui. For each pair i, j choose aﬃne open coverings

[ −1 −1 [ Ui ∩ Uj = Uijk and s (Ui) ∩ t (Uj) = Wijk. k∈Iij k∈Jij

Write Ui = Spec(Ai), Uijk = Spec(Aijk), Wijk = Spec(Bijk). Let κ be any inﬁnite cardinal ≥ than the cardinality of any of the sets I, Iij, Jij.

Let (F, α) be a quasi-coherent module on (U, R, s, t, c). Set Mi = F(Ui), Mijk = F(Uijk). Note that

Mi ⊗Ai Aijk = Mijk = Mj ⊗Aj Aijk and that α gives isomorphisms

α|Wijk : Mi ⊗Ai,t Bijk −→ Mj ⊗Aj ,s Bijk see Schemes, Lemma 7.3. Using the axiom of choice we choose a map (i, j, k, m) 7→ S(i, j, k, m) which associates to every i, j ∈ I, k ∈ Iij or k ∈ Jij and m ∈ Mi a ﬁnite subset S(i, j, k, m) ⊂ Mj such that we have

X 0 X 0 m ⊗ 1 = m ⊗ am0 or α(m ⊗ 1) = m ⊗ bm0 m0∈S(i,j,k,m) m0∈S(i,j,k,m) in Mijk for some am0 ∈ Aijk or bm0 ∈ Bijk. Moreover, let’s agree that S(i, i, k, m) = {m} for all i, j = i, k, m when k ∈ Iij. Fix such a collection S(i, j, k, m)

Given a family S = (Si)i∈I of subsets Si ⊂ Mi of cardinality at most κ we set 0 0 S = (Si) where 0 [ Sj = S(i, j, k, m) (i,j,k,m) such that m∈Si 0 0 Note that Si ⊂ Si. Note that Si has cardinality at most κ because it is a union over a set of cardinality at most κ of ﬁnite sets. Set S(0) = S, S(1) = S0 and by (n+1) (n) 0 (∞) S (n) (∞) (∞) induction S = (S ) . Then set S = n≥0 S . Writing S = (Si ) (∞) we see that for any element m ∈ Si the image of m in Mijk can be written as a P 0 0 (∞) ﬁnite sum m ⊗ am0 with m ∈ Sj . In this way we see that setting (∞) Ni = Ai-submodule of Mi generated by Si we have

Ni ⊗Ai Aijk = Nj ⊗Aj Aijk and α(Ni ⊗Ai,t Bijk) = Nj ⊗Aj ,s Bijk as submodules of Mijk or Mj ⊗Aj ,s Bijk. Thus there exists a quasi-coherent sub- ∗ ∗ ∗ module G ⊂ F with G(Ui) = Ni such that α(t G) = s G as submodules of s F. In other words, (G, α|t∗G) is a submodule of (F, α). Moreover, by construction G is κ-generated.

Let {(Gt, αt)}t∈T be the set of κ-generated quasi-coherent submodules of (F, α). 0 If t, t ∈ T then Gt + Gt0 is also a κ-generated quasi-coherent submodule as it is the image of the map Gt ⊕ Gt0 → F. Hence the system (ordered by inclusion) is directed. The arguments above show that every section of F over Ui is in one of the Gt (because we can start with S such that the given section is an element of Si). Hence colimt Gt → F is both injective and surjective as desired. GROUPOID SCHEMES 34

16. Groupoids and group schemes 03LK There are many ways to construct a groupoid out of an action a of a group G on a set V . We choose the one where we think of an element g ∈ G as an arrow with source v and target a(g, v). This leads to the following construction for group actions of schemes. 0234 Lemma 16.1. Let S be a scheme. Let Y be a scheme over S. Let (G, m) be a group scheme over Y with identity eG and inverse iG. Let X/Y be a scheme over Y and let a : G ×Y X → X be an action of G on X/Y . Then we get a groupoid scheme (U, R, s, t, c, e, i) over S in the following manner:

(1) We set U = X, and R = G ×Y X. (2) We set s : R → U equal to (g, x) 7→ x. (3) We set t : R → U equal to (g, x) 7→ a(g, x). 0 0 0 0 (4) We set c : R ×s,U,t R → R equal to ((g, x), (g , x )) 7→ (m(g, g ), x ). (5) We set e : U → R equal to x 7→ (eG(x), x). (6) We set i : R → R equal to (g, x) 7→ (iG(g), a(g, x)). Proof. Omitted. Hint: It is enough to show that this works on the set level. For this use the description above the lemma describing g as an arrow from v to a(g, v). 03LL Lemma 16.2. Let S be a scheme. Let Y be a scheme over S. Let (G, m) be a group scheme over Y . Let X be a scheme over Y and let a : G ×Y X → X be an action of G on X over Y . Let (U, R, s, t, c) be the groupoid scheme constructed in Lemma 16.1. The rule (F, α) 7→ (F, α) deﬁnes an equivalence of categories between G-equivariant OX -modules and the category of quasi-coherent modules on (U, R, s, t, c).

Proof. The assertion makes sense because t = a and s = pr1 as morphisms R = G ×Y X → X, see Definitions 12.1 and 14.1. Using the translation in Lemma 16.1 the commutativity requirements of the two definitions match up exactly. 17. The stabilizer group scheme 03LM Given a groupoid scheme we get a group scheme as follows. 0235 Lemma 17.1. Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. The scheme G defined by the cartesian square G / R

j=(t,s)

∆ U / U ×S U is a group scheme over U with composition law m induced by the composition law c. Proof. This is true because in a groupoid category the set of self maps of any object forms a group. −1 Since ∆ is an immersion we see that G = j (∆U/S) is a locally closed subscheme −1 of R. Thinking of it in this way, the structure morphism j (∆U/S) → U is induced by either s or t (it is the same), and m is induced by c. GROUPOID SCHEMES 35

0236Deﬁnition 17.2. Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. −1 The group scheme j (∆U/S) → U is called the stabilizer of the groupoid scheme (U, R, s, t, c). In the literature the stabilizer group scheme is often denoted S (because the word stabilizer starts with an “s” presumably); we cannot do this since we have already used S for the base scheme. 0237 Lemma 17.3. Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S, and let G/U be its stabilizer. Denote Rt/U the scheme R seen as a scheme over U via the morphism t : R → U. There is a canonical left action

a : G ×U Rt −→ Rt induced by the composition law c.

Proof. In terms of points over T/S we deﬁne a(g, r) = c(g, r). 04Q2 Lemma 17.4. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let G be the stabilizer group scheme of R. Let

G0 = G ×U,pr0 (U ×S U) = G ×S U

as a group scheme over U ×S U. The action of G on R of Lemma 17.3 induces an action of G0 on R over U ×S U which turns R into a pseudo G0-torsor over U ×S U.

Proof. This is true because in a groupoid category C the set MorC(x, y) is a principal homogeneous set under the group MorC(y, y). 04Q3 Lemma 17.5. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let p ∈ U ×S U be a point. Denote Rp the scheme theoretic ﬁbre of j = (t, s): R → U ×S U. If Rp 6= ∅, then the action

G0,κ(p) ×κ(p) Rp −→ Rp

(see Lemma 17.4) which turns Rp into a Gκ(p)-torsor over κ(p). Proof. The action is a pseudo-torsor by the lemma cited in the statement. And if Rp is not the empty scheme, then {Rp → p} is an fpqc covering which trivializes the pseudo-torsor.

18. Restricting groupoids 02VA Consider a (usual) groupoid C = (Ob, Arrows, s, t, c). Suppose we have a map of sets g : Ob0 → Ob. Then we can construct a groupoid C0 = (Ob0, Arrows0, s0, t0, c0) by thinking of a morphism between elements x0, y0 of Ob0 as a morphism in C between g(x0), g(y0). In other words we set

0 0 0 Arrows = Ob ×g,Ob,t Arrows ×s,Ob,g Ob . with obvious choices for s0, t0, and c0. There is a canonical functor C0 → C which is fully faithful, but not necessarily essentially surjective. This groupoid C0 endowed with the functor C0 → C is called the restriction of the groupoid C to Ob0. GROUPOID SCHEMES 36

02VB Lemma 18.1. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let g : U 0 → U be a morphism of schemes. Consider the following diagram

s0 0 0 0 R / R ×s,U U */ U

g 0 0 s t U ×U,t R / R / U

t g % U 0 / U where all the squares are ﬁbre product squares. Then there is a canonical compo- 0 0 0 0 0 0 0 0 0 sition law c : R ×s0,U 0,t0 R → R such that (U ,R , s , t , c ) is a groupoid scheme over S and such that U 0 → U, R0 → R deﬁnes a morphism (U 0,R0, s0, t0, c0) → (U, R, s, t, c) of groupoid schemes over S. Moreover, for any scheme T over S the functor of groupoids (U 0(T ),R0(T ), s0, t0, c0) → (U(T ),R(T ), s, t, c) is the restriction (see above) of (U(T ),R(T ), s, t, c) via the map U 0(T ) → U(T ).

Proof. Omitted. 02VCDefinition 18.2. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let g : U 0 → U be a morphism of schemes. The morphism of groupoids (U 0,R0, s0, t0, c0) → (U, R, s, t, c) constructed in Lemma 18.1 is called the restriction 0 0 of (U, R, s, t, c) to U . We sometime use the notation R = R|U 0 in this case. 02VD Lemma 18.3. The notions of restricting groupoids and (pre-)equivalence relations defined in Definitions 18.2 and 3.3 agree via the constructions of Lemmas 13.2 and 13.3. Proof. What we are saying here is that R0 of Lemma 18.1 is also equal to 0 0 0 0 0 R = (U ×S U ) ×U×S U R −→ U ×S U In fact this might have been a clearer way to state that lemma. 04ML Lemma 18.4. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let g : U 0 → U be a morphism of schemes. Let (U 0,R0, s0, t0, c0) be the restriction of (U, R, s, t, c) via g. Let G be the stabilizer of (U, R, s, t, c) and let G0 be the stabilizer of (U 0,R0, s0, t0, c0). Then G0 is the base change of G by g, i.e., there is a canonical 0 0 identification G = U ×g,U G. Proof. Omitted. 19. Invariant subschemes 03LN In this section we discuss briefly the notion of an invariant subscheme. 03BCDefinition 19.1. Let (U, R, s, t, c) be a groupoid scheme over the base scheme S. (1) A subset W ⊂ U is set-theoretically R-invariant if t(s−1(W )) ⊂ W . (2) An open W ⊂ U is R-invariant if t(s−1(W )) ⊂ W . (3) A closed subscheme Z ⊂ U is called R-invariant if t−1(Z) = s−1(Z). Here we use the scheme theoretic inverse image, see Schemes, Definition 17.7. GROUPOID SCHEMES 37

(4) A monomorphism of schemes T → U is R-invariant if T ×U,t R = R ×s,U T as schemes over R. For subsets and open subschemes W ⊂ U the R-invariance is also equivalent to requiring that s−1(W ) = t−1(W ) as subsets of R. If W ⊂ U is an R-equivariant −1 −1 open subscheme then the restriction of R to W is just RW = s (W ) = t (W ). Similarly, if Z ⊂ U is an R-invariant closed subscheme, then the restriction of R to −1 −1 Z is just RZ = s (Z) = t (Z). 03LO Lemma 19.2. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. (1) For any subset W ⊂ U the subset t(s−1(W )) is set-theoretically R-invariant. (2) If s and t are open, then for every open W ⊂ U the open t(s−1(W )) is an R-invariant open subscheme. (3) If s and t are open and quasi-compact, then U has an open covering consisting of R-invariant quasi-compact open subschemes. Proof. Part (1) follows from Lemmas 3.4 and 13.2, namely, t(s−1(W )) is the set of points of U equivalent to a point of W . Next, assume s and t open and W ⊂ U open. Since s is open the set W 0 = t(s−1(W )) is an open subset of U. Finally, assume that s, t are both open and quasi-compact. Then, if W ⊂ U is a quasi- compact open, then also W 0 = t(s−1(W )) is a quasi-compact open, and invariant by the discussion above. Letting W range over all affine opens of U we see (3). 0APA Lemma 19.3. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume s and t quasi-compact and flat and U quasi-separated. Let W ⊂ U be quasi-compact open. Then t(s−1(W )) is an intersection of a nonempty family of quasi-compact open subsets of U. Proof. Note that s−1(W ) is quasi-compact open in R. As a continuous map t maps the quasi-compact subset s−1(W ) to a quasi-compact subset t(s−1(W )). As t is flat and s−1(W ) is closed under generalization, so is t(s−1(W )), see (Morphisms, Lemma 25.9 and Topology, Lemma 19.6). Pick a quasi-compact open W 0 ⊂ U containing t(s−1(W )). By Properties, Lemma 2.4 we see that W 0 is a spectral space (here we use that U is quasi-separated). Then the lemma follows from Topology, −1 0 Lemma 24.7 applied to t(s (W )) ⊂ W . 0APB Lemma 19.4. Assumptions and notation as in Lemma 19.3. There exists an R- invariant open V ⊂ U and a quasi-compact open W 0 such that W ⊂ V ⊂ W 0 ⊂ U. Proof. Set E = t(s−1(W )). Recall that E is set-theoretically R-invariant (Lemma 19.2). By Lemma 19.3 there exists a quasi-compact open W 0 containing E. Let Z = U \ W 0 and consider T = t(s−1(Z)). Observe that Z ⊂ T and that E ∩ T = ∅ because s−1(E) = t−1(E) is disjoint from s−1(Z). Since T is the image of the closed subset s−1(Z) ⊂ R under the quasi-compact morphism t : R → U we see that any point ξ in the closure T is the specialization of a point of T , see Morphisms, Lemma 6.5 (and Morphisms, Lemma 6.3 to see that the scheme theoretic image 0 0 is the closure of the image). Say ξ ξ with ξ ∈ T . Suppose that r ∈ R and 0 0 0 s(r) = ξ. Since s is flat we can find a specialization r r in R such that s(r ) = ξ 0 0 (Morphisms, Lemma 25.9). Then t(r ) t(r). We conclude that t(r ) ∈ T as T is set-theoretically invariant by Lemma 19.2. Thus T is a set-theoretically R-invariant closed subset and V = U \ T is the open we are looking for. It is contained in W 0 which finishes the proof. GROUPOID SCHEMES 38

20. Quotient sheaves 02VE Let τ ∈ {Zariski, etale,´ fppf, smooth, syntomic}. Let S be a scheme. Let j : R → U ×S U be a pre-relation over S. Say U, R, S are objects of a τ-site Schτ (see Topologies, Section 2). Then we can consider the functors

opp hU , hR :(Sch/S)τ −→ Sets. These are sheaves, see Descent, Lemma 10.7. The morphism j induces a map j : hR → hU × hU . For each object T ∈ Ob((Sch/S)τ ) we can take the equivalence relation ∼T generated by j(T ): R(T ) → U(T ) × U(T ) and consider the quotient. Hence we get a presheaf

opp (20.0.1)02VF (Sch/S)τ −→ Sets,T 7−→ U(T )/ ∼T

02VGDefinition 20.1. Let τ, S, and the pre-relation j : R → U ×S U be as above. In this setting the quotient sheaf U/R associated to j is the sheafification of the presheaf (20.0.1) in the τ-topology. If j : R → U ×S U comes from the action of a group scheme G/S on U as in Lemma 16.1 then we sometimes denote the quotient sheaf U/G. This means exactly that the diagram

h / h U/R R / U /

is a coequalizer diagram in the category of sheaves of sets on (Sch/S)τ . Using the Yoneda embedding we may view (Sch/S)τ as a full subcategory of sheaves on (Sch/S)τ and hence identify schemes with representable functors. Using this abuse of notation we will often depict the diagram above simply

s R / U / U/R t / We will mostly work with the fppf topology when considering quotient sheaves of groupoids/equivalence relations. 03BDDefinition 20.2. In the situation of Definition 20.1. We say that the pre-relation j has a representable quotient if the sheaf U/R is representable. We will say a groupoid (U, R, s, t, c) has a representable quotient if the quotient U/R with j = (t, s) is representable. The following lemma characterizes schemes M representing the quotient. It applies for example if τ = fppf, U → M is flat, of finite presentation and surjective, and ∼ R = U ×M U. 03C5 Lemma 20.3. In the situation of Definition 20.1. Assume there is a scheme M, and a morphism U → M such that (1) the morphism U → M equalizes s, t, (2) the morphism U → M induces a surjection of sheaves hU → hM in the τ-topology, and (3) the induced map (t, s): R → U ×M U induces a surjection of sheaves

hR → hU×M U in the τ-topology. In this case M represents the quotient sheaf U/R. GROUPOID SCHEMES 39

Proof. Condition (1) says that hU → hM factors through U/R. Condition (2) says that U/R → hM is surjective as a map of sheaves. Condition (3) says that U/R → hM is injective as a map of sheaves. Hence the lemma follows. The following lemma is wrong if we do not require j to be a pre-equivalence relation (but just a pre-relation say). 045Y Lemma 20.4. Let τ ∈ {Zariski, etale,´ fppf, smooth, syntomic}. Let S be a scheme. Let j : R → U ×S U be a pre-equivalence relation over S. Assume U, R, S are objects of a τ-site Schτ . For T ∈ Ob((Sch/S)τ ) and a, b ∈ U(T ) the following are equivalent: (1) a and b map to the same element of (U/R)(T ), and (2) there exists a τ-covering {fi : Ti → T } of T and morphisms ri : Ti → R such that a ◦ fi = s ◦ ri and b ◦ fi = t ◦ ri. In other words, in this case the map of τ-sheaves

hR −→ hU ×U/R hU is surjective. Proof. Omitted. Hint: The reason this works is that the presheaf (20.0.1) in this case is really given by T 7→ U(T )/j(R(T )) as j(R(T )) ⊂ U(T ) × U(T ) is an equivalence relation, see Deﬁnition 3.1. 045Z Lemma 20.5. Let τ ∈ {Zariski, etale,´ fppf, smooth, syntomic}. Let S be a 0 scheme. Let j : R → U ×S U be a pre-equivalence relation over S and g : U → U 0 0 0 0 a morphism of schemes over S. Let j : R → U ×S U be the restriction of j to 0 0 U . Assume U, U ,R,S are objects of a τ-site Schτ . The map of quotient sheaves U 0/R0 −→ U/R

is injective. If g defines a surjection hU 0 → hU of sheaves in the τ-topology (for example if {g : U 0 → U} is a τ-covering), then U 0/R0 → U/R is an isomorphism. Proof. Suppose ξ, ξ0 ∈ (U 0/R0)(T ) are sections which map to the same section of 0 U/R. Then we can find a τ-covering T = {Ti → T } of T such that ξ|Ti , ξ |Ti are 0 0 given by ai, ai ∈ U (Ti). By Lemma 20.4 and the axioms of a site we may after refining T assume there exist morphisms ri : Ti → R such that g ◦ ai = s ◦ ri, 0 0 0 0 g ◦ ai = t ◦ ri. Since by construction R = R ×U×S U (U ×S U ) we see that 0 0 0 (ri, (ai, ai)) ∈ R (Ti) and this shows that ai and ai define the same section of 0 0 0 U /R over Ti. By the sheaf condition this implies ξ = ξ . 0 0 If hU 0 → hU is a surjection of sheaves, then of course U /R → U/R is surjective 0 also. If {g : U → U} is a τ-covering, then the map of sheaves hU 0 → hU is surjective, see Sites, Lemma 12.4. Hence U 0/R0 → U/R is surjective also in this case. 02VH Lemma 20.6. Let τ ∈ {Zariski, etale,´ fppf, smooth, syntomic}. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Let g : U 0 → U a morphism of schemes over S. Let (U 0,R0, s0, t0, c0) be the restriction of (U, R, s, t, c) to U 0. 0 Assume U, U ,R,S are objects of a τ-site Schτ . The map of quotient sheaves U 0/R0 −→ U/R GROUPOID SCHEMES 40

is injective. If the composition

0 U ×g,U,t R / R /( U pr1 s deﬁnes a surjection of sheaves in the τ-topology then the map is bijective. This 0 0 holds for example if {h : U ×g,U,t R → U} is a τ-covering, or if U → U deﬁnes a surjection of sheaves in the τ-topology, or if {g : U 0 → U} is a covering in the τ-topology. Proof. Injectivity follows on combining Lemmas 13.2 and 20.5. To see surjectivity (see Sites, Section 11 for a characterization of surjective maps of sheaves) we argue as follows. Suppose that T is a scheme and σ ∈ U/R(T ). There exists a covering

{Ti → T } such that σ|Ti is the image of some element fi ∈ U(Ti). Hence we may assume that σ is the image of f ∈ U(T ). By the assumption that h is a surjection of 0 sheaves, we can ﬁnd a τ-covering {ϕi : Ti → T } and morphisms fi : Ti → U ×g,U,t R 0 0 0 0 such that f◦ϕi = h◦fi. Denote fi = pr0◦fi : Ti → U . Then we see that fi ∈ U (Ti) 0 0 maps to g ◦ fi ∈ U(Ti) and that g ◦ fi ∼Ti h ◦ fi = f ◦ ϕi notation as in (20.0.1). Namely, the element of R(Ti) giving the relation is pr1 ◦ fi. This means that the 0 0 restriction of σ to Ti is in the image of U /R (Ti) → U/R(Ti) as desired. If {h} is a τ-covering, then it induces a surjection of sheaves, see Sites, Lemma 12.4. If U 0 → U is surjective, then also h is surjective as s has a section (namely the neutral element e of the groupoid scheme). 07S3 Lemma 20.7. Let S be a scheme. Let f :(U, R, j) → (U 0,R0, j0) be a morphism between equivalence relations over S. Assume that R / R0 f s s0 f U / U 0 is cartesian. For any τ ∈ {Zariski, etale,´ fppf, smooth, syntomic} the diagram U / U/R

f U 0 / U 0/R0 is a fibre product square of τ-sheaves. Proof. By Lemma 20.4 the quotient sheaves have a simple description which we will use below without further mention. We first show that 0 U −→ U ×U 0/R0 U/R is injective. Namely, assume a, b ∈ U(T ) map to the same element on the right hand side. Then f(a) = f(b). After replacing T by the members of a τ-covering we may assume that there exists an r ∈ R(T ) such that a = s(r) and b = t(r). Then r0 = f(r) is a T -valued point of R0 with s0(r0) = t0(r0). Hence r0 = e0(f(a)) (where e0 is the identity of the groupoid scheme associated to j0, see Lemma 13.3). Because the first diagram of the lemma is cartesian this implies that r has to equal e(a). Thus a = b. GROUPOID SCHEMES 41

Finally, we show that the displayed arrow is surjective. Let T be a scheme over S 0 0 and let (a , b) be a section of the sheaf U ×U 0/R0 U/R over T . After replacing T by the members of a τ-covering we may assume that b is the class of an element b ∈ U(T ). After replacing T by the members of a τ-covering we may assume that there exists an r0 ∈ R0(T ) such that a0 = t(r0) and s0(r0) = f(b). Because the first diagram of the lemma is cartesian we can find r ∈ R(T ) such that s(r) = b and f(r) = r0. Then it is clear that a = t(r) ∈ U(T ) is a section which maps to 0 (a , b). 21. Descent in terms of groupoids 0APC Cartesian morphisms are defined as follows. 0APDDefinition 21.1. Let S be a scheme. Let f :(U 0,R0, s0, t0, c0) → (U, R, s, t, c) be a morphism of groupoid schemes over S. We say f is cartesian, or that (U 0,R0, s0, t0, c0) is cartesian over (U, R, s, t, c), if the diagram R0 / R f s0 s f U 0 / U is a fibre square in the category of schemes. A morphism of groupoid schemes cartesian over (U, R, s, t, c) is a morphism of groupoid schemes compatible with the structure morphisms towards (U, R, s, t, c). Cartesian morphisms are related to descent data. First we prove a general lemma describing the category of cartesian groupoid schemes over a fixed groupoid scheme. 0APE Lemma 21.2. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. The category of groupoid schemes cartesian over (U, R, s, t, c) is equivalent to the category of pairs (V, ϕ) where V is a scheme over U and

ϕ : V ×U,t R −→ R ×s,U V ∗ is an isomorphism over R such that e ϕ = idV and such that ∗ ∗ ∗ c ϕ = pr1ϕ ◦ pr0ϕ

as morphisms of schemes over R ×s,U,t R. Proof. The pullback notation in the lemma signiﬁes base change. The displayed formula makes sense because

(R ×s,U,t R) ×pr1,R,pr1 (V ×U,t R) = (R ×s,U,t R) ×pr0,R,pr0 (R ×s,U V )

as schemes over R ×s,U,t R. 0 0 0 0 0 Given (V, ϕ) we set U = V and R = V ×U,t R. We set t : R → U equal to 0 the projection V ×U,t R → V . We set s equal to ϕ followed by the projection 0 R ×s,U V → V . We set c equal to the composition

0 0 ϕ,1 R ×s0,U 0,t0 R −−→ (R ×s,U V ) ×V (V ×U,t R)

−→ R ×s,U V ×U,t R ϕ−1,1 −−−−→ V ×U,t (R ×s,U,t R)

1,c 0 −−→ V ×U,t R = R GROUPOID SCHEMES 42

A computation, which we omit shows that we obtain a groupoid scheme over (U, R, s, t, c). It is clear that this groupoid scheme is cartesian over (U, R, s, t, c). Conversely, given f :(U 0,R0, s0, t0, c0) → (U, R, s, t, c) cartesian then the morphisms

0 0 0 t ,f 0 f,s 0 U ×U,t R ←−− R −−→ R ×s,U U are isomorphisms and we can set V = U 0 and ϕ equal to the composition (f, s0) ◦ (t0, f)−1. We omit the proof that ϕ satisfies the conditions in the lemma. We omit the proof that these constructions are mutually inverse. Let S be a scheme. Let f : X → Y be a morphism of schemes over S. Then we obtain a groupoid scheme (X,X ×Y X, pr1, pr0, c) over S. Namely, j : X ×Y X → X ×S X is an equivalence relation and we can take the associated groupoid, see Lemma 13.3. 0APF Lemma 21.3. Let S be a scheme. Let f : X → Y be a morphism of schemes over S. The construction of Lemma 21.2 determines an equivalence category of groupoid schemes category of descent data −→ cartesian over (X,X ×Y X,...) relative to X/Y Proof. This is clear from Lemma 21.2 and the definition of descent data on schemes in Descent, Definition 31.1. 22. Separation conditions

02YG This really means conditions on the morphism j : R → U ×S U when given a groupoid (U, R, s, t, c) over S. As in the previous section we ﬁrst formulate the corresponding diagram. 02YH Lemma 22.1. Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. Let G → U be the stabilizer group scheme. The commutative diagram

R / R ×s,U U / U f7→(f,s(f)) ∆ R/U×S U (f,g)7→(f,f −1◦g) R ×(U×S U) R / R ×s,U G / G the two left horizontal arrows are isomorphisms and the right square is a ﬁbre product square. Proof. Omitted. Exercise in the deﬁnitions and the functorial point of view in algebraic geometry. 02YI Lemma 22.2. Let S be a scheme. Let (U, R, s, t, c) be a groupoid over S. Let G → U be the stabilizer group scheme. (1) The following are equivalent (a) j : R → U ×S U is separated, (b) G → U is separated, and (c) e : U → G is a closed immersion. (2) The following are equivalent (a) j : R → U ×S U is quasi-separated, (b) G → U is quasi-separated, and (c) e : U → G is quasi-compact. GROUPOID SCHEMES 43

Proof. The group scheme G → U is the base change of R → U ×S U by the diagonal morphism U → U ×S U, see Lemma 17.1. Hence if j is separated (resp. quasi-separated), then G → U is separated (resp. quasi-separated). (See Schemes, Lemma 21.12). Thus (a) ⇒ (b) in both (1) and (2). If G → U is separated (resp. quasi-separated), then the morphism U → G, as a section of the structure morphism G → U is a closed immersion (resp. quasi- compact), see Schemes, Lemma 21.11. Thus (b) ⇒ (a) in both (1) and (2). By the result of Lemma 22.1 (and Schemes, Lemmas 18.2 and 19.3) we see that if e

is a closed immersion (resp. quasi-compact) ∆R/U×S U is a closed immersion (resp. quasi-compact). Thus (c) ⇒ (a) in both (1) and (2). 23. Finite flat groupoids, affine case 03BE Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(A), and R = Spec(B) are affine. In this case we get two ring maps s], t] : A −→ B. Let C be the equalizer of s] and t]. In a formula (23.0.1)03BF C = {a ∈ A | t](a) = s](a)}. We will sometimes call this the ring of R-invariant functions on U. What properties does M = Spec(C) have? The first observation is that the diagram

R s / U

t U / M is commutative, i.e., the morphism U → M equalizes s, t. Moreover, if T is any affine scheme, and if U → T is a morphism which equalizes s, t, then U → T factors through U → M. In other words, U → M is a coequalizer in the category of affine schemes. We would like to find conditions that guarantee the morphism U → M is really a “quotient” in the category of schemes. We will discuss this at length elsewhere (insert future reference here); here we just discuss some special cases. Namely, we will focus on the case where s, t are finite locally free.

03BGExample 23.1. Let k be a field. Let U = GL2,k. Let B ⊂ GL2 be the closed subgroup scheme of upper triangular matrices. Then the quotient sheaf GL2,k/B (in the Zariski, étale or fppf topology, see Definition 20.1) is representable by the 1 projective line: P = GL2,k/B. (Details omitted.) On the other hand, the ring of invariant functions in this case is just k. Note that in this case the morphisms s, t : R = GL2,k ×k B → GL2,k = U are smooth of relative dimension 3. Recall that in Exercises, Exercises 22.6 and 22.7 we have defined the determinant and the norm for finitely locally free modules and finite locally free ring extensions. If ϕ : A → B is a finite locally free ring map, then we will denote Normϕ(b) ∈ A the norm of b ∈ B. In the case of a finite locally free morphism of schemes, the norm was constructed in Divisors, Lemma 17.6. 03BH Lemma 23.2. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(A) and R = Spec(B) are affine and s, t : R → U finite locally ] free. Let C be as in (23.0.1). Let f ∈ A. Then Norms] (t (f)) ∈ C. GROUPOID SCHEMES 44

Proof. Consider the commutative diagram U : d t t

R o R ×s,U,t R / R pr0 c

s pr1 s

t s URo / U

of Lemma 13.4. Think of f ∈ Γ(U, OU ). The commutativity of the top part of the ] ] ] ] diagram shows that pr0(t (f)) = c (t (f)) as elements of Γ(R ×S,U,t R, O). Looking at the right lower cartesian square the compatibility of the norm construction with ] ] ] ] base change shows that s (Norms] (t (f))) = Norm ] (c (t (f))). Similarly we get pr1 ] ] ] ] t (Norms] (t (f))) = Norm ] (pr0(t (f))). Hence by the ﬁrst equality of this proof pr1 ] ] ] ] we see that s (Norms] (t (f))) = t (Norms] (t (f))) as desired. 03BI Lemma 23.3. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume s, t : R → U ﬁnite locally free. Then a U = Ur r≥1

is a disjoint union of R-invariant opens such that the restriction Rr of R to Ur has the property that s, t : Rr → Ur are finite locally free of rank r. ` Proof. By Morphisms, Lemma 48.5 there exists a decomposition U = r≥0 Ur −1 such that s : s (Ur) → Ur is finite locally free of rank r. As s is surjective we see −1 that U0 = ∅. Note that u ∈ Ur ⇔ if and only if the scheme theoretic fibre s (u) has degree r over κ(u). Now, if z ∈ R with s(z) = u and t(z) = u0 then using notation as in Lemma 13.4 −1 pr1 (z) → Spec(κ(z)) is the base change of both s−1(u) → Spec(κ(u)) and s−1(u0) → Spec(κ(u0)) by the 0 lemma cited. Hence u ∈ Ur ⇔ u ∈ Ur, in other words, the open subsets Ur are −1 R-invariant. In particular the restriction of R to Ur is just s (Ur) and s : Rr → Ur is finite locally free of rank r. As t : Rr → Ur is isomorphic to s by the inverse of Rr we see that it has also rank r. 03BJ Lemma 23.4. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(A) and R = Spec(B) are affine and s, t : R → U finite locally free. Let C ⊂ A be as in (23.0.1). Then A is integral over C. Proof. First, by Lemma 23.3 we know that (U, R, s, t, c) is a disjoint union of groupoid schemes (Ur,Rr, s, t, c) such that each s, t : Rr → Ur has constant rank r. As U is quasi-compact, we have Ur = ∅ for almost all r. It suffices to prove the lemma for each (Ur,Rr, s, t, c) and hence we may assume that s, t are finite locally free of rank r. Assume that s, t are finite locally free of rank r. Let f ∈ A. Consider the element x − f ∈ A[x], where we think of x as the coordinate on A1. Since 1 1 (U × A ,R × A , s × idA1 , t × idA1 , c × idA1 ) GROUPOID SCHEMES 45

is also a groupoid scheme with finite source and target, we may apply Lemma 23.2 ] to it and we see that P (x) = Norms] (t (x − f)) is an element of C[x]. Because s] : A → B is finite locally free of rank r we see that P is monic of degree r. Moreover P (f) = 0 by Cayley-Hamilton (Algebra, Lemma 16.1). 03BK Lemma 23.5. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(A) and R = Spec(B) are affine and s, t : R → U finite locally free. Let C ⊂ A be as in (23.0.1). Let C → C0 be a ring map, and set 0 0 0 0 U = Spec(A ⊗C C ), R = Spec(B ⊗C C ). Then (1) The maps s, t, c induce maps s0, t0, c0 such that (U 0,R0, s0, t0, c0) is a groupoid scheme. Let C1 ⊂ A0 be the R0-invariant functions on U 0. (2) The canonical map ϕ : C0 → C1 satisfies (a) for every f ∈ C1 there exists an n > 0 and a polynomial P ∈ C0[x] whose image in C1[x] is (x − f)n, and (b) for every f ∈ Ker(ϕ) there exists an n > 0 such that f n = 0. (3) If C → C0 is flat then ϕ is an isomorphism.

0 0 0 Proof. The proof of part (1) is omitted. Let us denote A = A ⊗C C and B = 0 B ⊗C C . Then we have 1 0 0 ] 0 ] 0 ] ] C = {a ∈ A | (t ) (a) = (s ) (a)} = {a ∈ A ⊗C C | t ⊗ 1(a) = s ⊗ 1(a)}. In other words, C1 is the kernel of the difference map (t] − s]) ⊗ 1 which is just the base change of the C-linear map t] − s] : A → B by C → C0. Hence (3) follows. Proof of part (2)(b). Since C → A is integral (Lemma 23.4) and injective we see that Spec(A) → Spec(C) is surjective, see Algebra, Lemma 36.17. Thus also Spec(A0) → Spec(C0) is surjective as a base change of a surjective morphism (Mor- phisms, Lemma 9.4). Hence Spec(C1) → Spec(C0) is surjective also. This implies (2)(b) holds for example by Algebra, Lemma 30.6. Proof of part (2)(a). By Lemma 23.3 our groupoid scheme (U, R, s, t, c) decomposes as a finite disjoint union of groupoid schemes (Ur,Rr, s, t, c) such that s, t : Rr → Ur are finite locally free of rank r. Pulling back by U 0 = Spec(C0) → U we obtain a similar decomposition of U 0 and U 1 = Spec(C1). We will show in the next 0 1 paragraph that (2)(a) holds for the corresponding system of rings Ar,Br,Cr,Cr,Cr 1 with n = r. Then given f ∈ C let Pr ∈ Cr[x] be the polynomial whose image in 1 r Cr [x] is the image of (x − f) . Choosing a sufficiently divisible integer n we see that there is a polynomial P ∈ C0[x] whose image in C1[x] is (x − f)n; namely, we 0 0 n/r take P to be the unique element of C [x] whose image in Cr[x] is Pr . In this paragraph we prove (2)(a) in case the ring maps s], t] : A → B are finite 1 0 0 locally free of a fixed rank r. Let f ∈ C ⊂ A = A ⊗C C . Choose a flat C- 0 algebra D and a surjection D → C . Choose a lift g ∈ A ⊗C D of f. Consider the polynomial ] P = Norms]⊗1(t ⊗ 1(x − g))

in (A ⊗C D)[x]. By Lemma 23.2 and part (3) of the current lemma the coeﬃcients of P are in D (compare with the proof of Lemma 23.4). On the other hand, the 0 r ] ] ] image of P in (A ⊗C C )[x] is (x − f) because t ⊗ 1(x − f) = s (x − f) and s is ﬁnite locally free of rank r. This proves what we want with P as in the statement 0 (2)(a) given by the image of our P under the map D[x] → C [x]. GROUPOID SCHEMES 46

03BL Lemma 23.6. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume U = Spec(A) and R = Spec(B) are aﬃne and s, t : R → U ﬁnite locally free. Let C ⊂ A be as in (23.0.1). Then U → M = Spec(C) has the following properties:

(1) the map on points |U| → |M| is surjective and u0, u1 ∈ |U| map to the same point if and only if there exists a r ∈ |R| with t(r) = u0 and s(r) = u1, in a formula |M| = |U|/|R| (2) for any algebraically closed field k we have M(k) = U(k)/R(k) Proof. Since C → A is integral (Lemma 23.4) and injective we see that Spec(A) → Spec(C) is surjective, see Algebra, Lemma 36.17. Thus |U| → |M| is surjective. Let k be an algebraically closed field and let C → k be a ring map. Since surjective morphisms are preserved under base change (Morphisms, Lemma 9.4) we see that A ⊗C k is not zero. Now k ⊂ A ⊗C k is a nonzero integral extension. Hence any residue field of A ⊗C k is an algebraic extension of k, hence equal to k. Thus we see that U(k) → M(k) is surjective.

Let a0, a1 : A → k be two ring maps. If there exists a ring map b : B → k such ] ] that a0 = b ◦ t and a1 = b ◦ s then we see that a0|C = a1|C by deﬁnition. Thus the map U(k) → M(k) equalizes the two maps R(k) → U(k). Conversely, suppose that a0|C = a1|C . Let us name this algebra map c : C → k. Consider the diagram B O O

a x 1 koo A f a0 O

c C If we can construct a dotted arrow making the diagram commute, then the proof of part (2) of the lemma is complete. Since s : A → B is finite there exist finitely ] many ring maps b1, . . . , bn : B → k such that bi ◦ s = a1. If the dotted arrow does 0 ] not exist, then we see that none of the ai = bi ◦t , i = 1, . . . , n is equal to a0. Hence the maximal ideals 0 0 mi = Ker(ai ⊗ 1 : A ⊗C k → k) of A ⊗C k are distinct from m = Ker(a0 ⊗ 1 : A ⊗C k → k). By Algebra, Lemma 0 15.2 we would get an element f ∈ A ⊗C k with f ∈ m, but f 6∈ mi for i = 1, . . . , n. Consider the norm ] g = Norms]⊗1(t ⊗ 1(f)) ∈ A ⊗C k 1 By Lemma 23.2 this lies in the invariants C ⊂ A ⊗C k of the base change groupoid ∗ (base change via the map c : C → k). On the one hand, a1(g) ∈ k since the value ] of t (f) at all the points (which correspond to b1, . . . , bn) lying over a1 is invertible (insert future reference on property determinant here). On the other hand, since f ∈ m, we see that f is not a unit, hence t](f) is not a unit (as t] ⊗ 1 is faithfully flat), hence its norm is not a unit (insert future reference on property determinant here). We conclude that C1 contains an element which is not nilpotent and not a GROUPOID SCHEMES 47

unit. We will now show that this leads to a contradiction. Namely, apply Lemma 23.5 to the map c : C → C0 = k, then we see that the map of k into the invariants C1 is injective and moreover, that for any element x ∈ C1 there exists an integer n > 0 such that xn ∈ k. Hence every element of C1 is either a unit or nilpotent. We still have to finish the proof of (1). We already know that |U| → |M| is surjective. It is clear that |U| → |M| is |R|-invariant. Finally, suppose u0, u1 ∈ U maps to the same point m ∈ M. Then the induced field extensions κ(u0)/κ(m) and κ(u1)/κ(m) are algebraic (as A is integral over C as used above). Hence if k is an algebraic closure of κ(m) then we can find κ(m)-embeddings u0 : κ(u0) → k and u1 : κ(u1) → k. These determine k-valued points u0, u1 ∈ U(k) mapping to the same point of M(k). By part (2) we see that there exists a point r ∈ R(k) with s(r) = u0 and t(r) = u1. The image r ∈ R of r is a point with s(r) = u0 and t(r) = u1 as desired. 0DT9 Lemma 23.7. Let S be a scheme. Let f :(U 0,R0, s0, t0) → (U, R, s, t, c) be a morphism of groupoid schemes over S. (1) U, R, U 0, R0 are affine, (2) s, t, s0, t0 are finite locally free, (3) the diagrams

R0 / R R0 / R G0 / G f f f s0 s t0 t f f f U 0 / U U 0 / U U 0 / U are cartesian where G and G0 are the stabilizer group schemes, and (4) f : U 0 → U is étale. Then the map C → C0 from the R-invariant functions on U to the R0-invariant 0 0 0 functions on U is étale and U = Spec(C ) ×Spec(C) U. Proof. Set M = Spec(C) and M 0 = Spec(C0). Write U = Spec(A), U 0 = Spec(A0), R = Spec(B), and R0 = Spec(B0). We will use the results of Lemmas 23.4, 23.5, and 23.6 without further mention. Assume C is a strictly henselian local ring. Let p ∈ M be the closed point and let p0 ∈ M 0 map to p. Claim: in this case there is a disjoint union decomposition (U 0,R0, s0, t0, c0) = (U, R, s, t, c) q (U 00,R00, s00, t00, c00) over (U, R, s, t, c) such that for the corresponding disjoint union decomposition M 0 = M q M 00 over M the point p0 corresponds to p ∈ M.

The claim implies the lemma. Suppose that M1 → M is a flat morphism of affine schemes. Then we can base change everything to M1 without affecting the hy- 0 potheses (1) – (4). From Lemma 23.5 we see M1, resp. M1 is the spectrum of the 0 0 R1-invariant functions on U1, resp. the R1-invariant functions on U1. Suppose that 0 0 p ∈ M maps to p ∈ M. Let M1 be the spectrum of the strict henselization of 0 0 0 OM,p with closed point p1 ∈ M1. Choose a point p1 ∈ M1 mapping to p1 and p . From the claim we get 0 0 0 0 0 00 00 00 00 00 (U1,R1, s1, t1, c1) = (U1,R1, s1, t1, c1) q (U1 ,R1 , s1 , t1 , c1 ) 0 00 and correspondingly M1 = M1 q M1 as a scheme over M1. Write M1 = Spec(C1) and write C1 = colim Ci as a filtered colimit of étale C-algebras. Set Mi = Spec(Ci). GROUPOID SCHEMES 48

The M1 = lim Mi and similarly for the other schemes. By Limits, Lemmas 4.11 and 8.11 we can find an i such that 0 0 0 0 0 00 00 00 00 00 (Ui ,Ri, si, ti, ci) = (Ui,Ri, si, ti, ci) q (Ui ,Ri , si , ti , ci ) 0 00 0 We conclude that Mi = Mi q Mi . In particular M → M becomes étale at a point over p0 after an étale base change. This implies that M 0 → M is étale at p0 (for 0 ∼ 0 example by Morphisms, Lemma 36.17). We will prove U = M ×M U after we prove the claim. 0 Proof of the claim. Observe that Up and Up0 have finitely many points. For u ∈ Up we have κ(u)/κ(p) is algebraic, hence κ(u) is separably closed. As U 0 → U is 0 étale, we conclude the morphism Up0 → Up induces isomorphisms on residue field 0 0 extensions. Let u ∈ Up0 with image u ∈ Up. By assumption (3) the morphism of 0 −1 0 −1 0 −1 0 −1 0 scheme theoretic fibres (s ) (u ) → s (u), (t ) (u ) → t (u), and Gu0 → Gu −1 are isomorphisms. Observing that Up = t(s (u)) (set theoretically) we conclude 0 0 0 that the points of Up0 surject onto the points of Up. Suppose that u1 and u2 are 0 0 0 points of Up0 mapping to the same point u of Up. Then there exists a point r ∈ Rp0 0 0 0 0 0 0 with s (r ) = u1 and t (r ) = u2. Consider the two towers of fields 0 0 0 0 κ(r )/κ(u1)/κ(u)/κ(p) κ(r )/κ(u2)/κ(u)/κ(p) whose “ends” are the same as the two “ends” of the two towers 0 0 0 0 0 0 κ(r )/κ(u1)/κ(p )/κ(p) κ(r )/κ(u2)/κ(p )/κ(p) 0 0 0 0 0 0 0 These two induce the same maps κ(p ) → κ(r ) as (Up0 ,Rp0 , s , t , c ) is a groupoid over p0. Since κ(u)/κ(p) is purely inseparable, we conclude that the two induced 0 0 maps κ(u) → κ(r ) are the same. Therefore r maps to a point of the fibre Gu. 0 0 By assumption (3) we conclude that r ∈ (G ) 0 . Namely, we may think of G as u1 a closed subscheme of R viewed as a scheme over U via s and use that the base 0 0 0 0 0 change to U gives G ⊂ R . In particular we have u1 = u2. We conclude that 0 Up0 → Up is a bijective map on points inducing isomorphisms on residue fields. It 0 0 follows that Up0 is a finite set of closed points (Algebra, Lemma 35.9) and hence Up0 0 0 0 0 is closed in U . Let J ⊂ A be the radical ideal cutting out Up0 set theoretically. Second part proof of the claim. Let m ⊂ C be the maximal ideal. Observe√ that (A, mA) is a henselian pair by More on Algebra, Lemma 11.8. Let J = mA. Then (A, J) is a henselian pair (More on Algebra, Lemma 11.7) and the étale ring map A → A0 induces an isomorphism A/J → A0/J 0 by our deliberations above. We conclude that A0 = A × A00 by More on Algebra, Lemma 11.6. Consider the corresponding disjoint union decomposition U 0 = U q U 00. The open (s0)−1(U) 0 0 0 −1 is the set of points of R specializing to a point of Rp0 . Similarly for (t ) (U). Similarly we have (s0)−1(U 00) = (t0)−1(U 00) as this is the set of points which do not 0 specialize to Rp0 . Hence we obtain a disjoint union decomposition (U 0,R0, s0, t0, c0) = (U, R, s, t, c) q (U 00,R00, s00, t00, c00) This immediately gives M 0 = M q M 00 and the proof of the claim is complete. 0 0 We still have to prove that the canonical map U → M ×M U is an isomorphism. It is an étale morphism (Morphisms, Lemma 36.18). On the other hand, by base changing to strictly henselian local rings (as in the third paragraph of the proof) and 0 using the bijectivity Up0 → Up established in the course of the proof of the claim, we 0 0 see that U → M ×M U is universally bijective (some details omitted). However, a GROUPOID SCHEMES 49

universally bijective étale morphism is an isomorphism (Descent, Lemma 22.2) and the proof is complete. 03C8 Lemma 23.8. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume (1) U = Spec(A), and R = Spec(B) are aﬃne, and L ] ] (2) there exist elements xi ∈ A, i ∈ I such that B = i∈I s (A)t (xi). L ∼ Then A = i∈I Cxi, and B = A ⊗C A where C ⊂ A is the R-invariant functions on U as in (23.0.1). Proof. During this proof we will write s, t : A → B instead of s], t], and similarly c : B → B ⊗s,A,t B. We write p0 : B → B ⊗s,A,t B, b 7→ b ⊗ 1 and p1 : B → B ⊗s,A,t B, b 7→ 1 ⊗ b. By Lemma 13.5 and the deﬁnition of C we have the following commutative diagram c o B ⊗s,A,t BB o A o t O p0 O O p1 s s B o A o C o t Moreover the tow left squares are cocartesian in the category of rings, and the top row is isomorphic to the diagram

p1 o B ⊗t,A,t BB o A o t p0 which is an equalizer diagram according to Descent, Lemma 3.6 because condition (2) implies in particular that s (and hence also then isomorphic arrow t) is faithfully ﬂat. The lower row is an equalizer diagram by deﬁnition of C. We can use the xi and get a commutative diagram c o B ⊗s,A,t BB o A o t O p0 O O p1 s s L o L L i∈I Bxi i∈I Axi o i∈I Cxi o t

where in the right vertical arrow we map xi to xi, in the middle vertical arrow we map xi to t(xi) and in the left vertical arrow we map xi to c(t(xi)) = t(xi) ⊗ 1 = p0(t(xi)) (equality by the commutativity of the top part of the diagram in Lemma 13.4). Then the diagram commutes. Moreover the middle vertical arrow is an isomorphism by assumption. Since the left two squares are cocartesian we conclude that also the left vertical arrow is an isomorphism. On the other hand, the horizontal rows are exact (i.e., they are equalizers). Hence we conclude that also the right vertical arrow is an isomorphism. 03BM Proposition 23.9. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume (1) U = Spec(A), and R = Spec(B) are aﬃne, (2) s, t : R → U ﬁnite locally free, and (3) j = (t, s) is an equivalence. GROUPOID SCHEMES 50

In this case, let C ⊂ A be as in (23.0.1). Then U → M = Spec(C) is finite locally free and R = U ×M U. Moreover, M represents the quotient sheaf U/R in the fppf topology (see Definition 20.1). Proof. During this proof we use the notation s, t : A → B instead of the notation s], t]. By Lemma 20.3 it suffices to show that C → A is finite locally free and that the map t ⊗ s : A ⊗C A −→ B is an isomorphism. First, note that j is a monomorphism, and also finite (since already s and t are finite). Hence we see that j is a closed immersion by Morphisms, Lemma 44.15. Hence A ⊗C A → B is surjective. We will perform base change by flat ring maps C → C0 as in Lemma 23.5, and we will use that formation of invariants commutes with flat base change, see part (3) of the lemma cited. We will show below that for every prime p ⊂ C, there exists a 0 0 local flat ring map Cp → Cp such that the result holds after a base change to Cp. This implies immediately that A⊗C A → B is injective (use Algebra, Lemma 23.1). It also implies that C → A is flat, by combining Algebra, Lemmas 39.17, 39.18, and 39.8. Then since U → Spec(C) is surjective also (Lemma 23.6) we conclude ∼ that C → A is faithfully flat. Then the isomorphism B = A ⊗C A implies that A is a finitely presented C-module, see Algebra, Lemma 83.2. Hence A is finite locally free over C, see Algebra, Lemma 78.2.

By Lemma 23.3 we know that A is a finite product of rings Ar and B is a finite product of rings Br such that the groupoid scheme decomposes accordingly (see the proof of Lemma 23.4). Then also C is a product of rings Cr and correspondingly C0 decomposes as a product. Hence we may and do assume that the ring maps s, t : A → B are finite locally free of a fixed rank r. 0 The local ring maps Cp → Cp we are going to use are any local flat ring maps such 0 that the residue field of Cp is infinite. By Algebra, Lemma 159.1 such local ring maps exist. Assume C is a local ring with maximal ideal m and infinite residue field, and assume that s, t : A → B is finite locally free of constant rank r > 0. Since C ⊂ A is integral (Lemma 23.4) all primes lying over m are maximal, and all maximal ideals of A lie over m. Similarly for C ⊂ B. Pick a maximal ideal m0 of A lying over m (exists by Lemma 23.6). Since t : A → B is finite locally free there exist at most finitely many maximal ideals of B lying over m0. Hence we conclude (by Lemma 23.6 again) that A has finitely many maximal ideals, i.e., A is semi-local. This in turn implies that B is semi-local as well. OK, and now, because t⊗s : A⊗C A → B is surjective, we can apply Algebra, Lemma 78.8 to the ring map C → A, the A-module M = B (seen as an A-module via t) and the C-submodule s(A) ⊂ B. This lemma implies that there exist x1, . . . , xr ∈ A such that M is free over A on the basis s(x1), . . . , s(xr). ∼ Hence we conclude that C → A is finite free and B = A ⊗C A by applying Lemma 23.8. 24. Finite flat groupoids 03JD In this section we prove a lemma that will help to show that the quotient of a scheme by a finite flat equivalence relation is a scheme, provided that each equivalence class is contained in an affine. See Properties of Spaces, Proposition 14.1. GROUPOID SCHEMES 51

03JE Lemma 24.1. Let S be a scheme. Let (U, R, s, t, c) be a groupoid scheme over S. Assume s, t are finite locally free. Let u ∈ U be a point such that t(s−1({u})) is contained in an affine open of U. Then there exists an R-invariant affine open neighbourhood of u in U. Proof. Since s is finite locally free it has finite fibres. Hence t(s−1({u})) = {u1, . . . , un} is a finite set. Note that u ∈ {u1, . . . , un}. Let W ⊂ U be an affine −1 open containing {u1, . . . , un}, in particular u ∈ W . Consider Z = R \ s (W ) ∩ t−1(W ). This is a closed subset of R. The image t(Z) is a closed subset of U which can be loosely described as the set of points of U which are R-equivalent to a point of U \ W . Hence W 0 = U \ t(Z) is an R-invariant, open subscheme of U contained 0 in W , and {u1, . . . , un} ⊂ W . Picture 0 {u1, . . . , un} ⊂ W ⊂ W ⊂ U. 0 Let f ∈ Γ(W, OW ) be an element such that {u1, . . . , un} ⊂ D(f) ⊂ W . Such an f exists by Algebra, Lemma 15.2. By our choice of W 0 we have s−1(W 0) ⊂ t−1(W ), and hence we get a diagram s−1(W 0) W t / s W 0 The vertical arrow is finite locally free by assumption. Set ] 0 g = Norms(t f) ∈ Γ(W , OW 0 ) By construction g is a function on W 0 which is nonzero in u, as t](f) is nonzero in each of the points of R lying over u, since f is nonzero in u1, . . . , un. Similarly, D(g) ⊂ W 0 is equal to the set of points w such that f is not zero in any of the points equivalent to w. This means that D(g) is an R-invariant affine open of W 0. The final picture is 0 {u1, . . . , un} ⊂ D(g) ⊂ D(f) ⊂ W ⊂ W ⊂ U and hence we win. 25. Descending quasi-projective schemes 0CCH We can use Lemma 24.1 to show that a certain type of descent datum is effective. 0CCI Lemma 25.1. Let X → Y be a surjective finite locally free morphism. Let V be a scheme over X such that for all (y, v1, . . . , vd) where y ∈ Y and v1, . . . , vd ∈ Vy there exists an affine open U ⊂ V with v1, . . . , vd ∈ U. Then any descent datum on V/X/Y is effective. Proof. Let ϕ be a descent datum as in Descent, Definition 31.1. Recall that the functor from schemes over Y to descent data relative to {X → Y } is fully faithful, see Descent, Lemma 32.11. Thus using Constructions, Lemma 2.1 it suffices to prove the lemma in the case that Y is affine. Some details omitted (this argument can be avoided if Y is separated or has affine diagonal, because then every morphism from an affine scheme to X is affine). Assume Y is affine. If V is also affine, then we have effectivity by Descent, Lemma 34.1. Hence by Descent, Lemma 32.13 it suffices to prove that every GROUPOID SCHEMES 52

point v of V has a ϕ-invariant aﬃne open neighbourhood. Consider the groupoid (X,X ×Y X, pr1, pr0, pr02). By Lemma 21.3 the descent datum ϕ determines and is determined by a cartesian morphism of groupoid schemes

(V, R, s, t, c) −→ (X,X ×Y X, pr1, pr0, pr02)

over Spec(Z). Since X → Y is finite locally free, we see that pri : X ×Y X → X and hence s and t are finite locally free. In particular the R-orbit t(s−1({v})) of our point v ∈ V is finite. Using the equivalence of categories of Lemma 21.3 once more we see that ϕ-invariant opens of V are the same thing as R-invariant opens of V . Our assumption shows there exists an affine open of V containing the orbit t(s−1({v})) as all the points in this orbit map to the same point of Y . Thus Lemma 24.1 provides an R-invariant affine open containing v. 0CCJ Lemma 25.2. Let X → Y be a surjective finite locally free morphism. Let V be a scheme over X such that one of the following holds (1) V → X is projective, (2) V → X is quasi-projective, (3) there exists an ample invertible sheaf on V , (4) there exists an X-ample invertible sheaf on V , (5) there exists an X-very ample invertible sheaf on V . Then any descent datum on V/X/Y is effective.

Proof. We check the condition in Lemma 25.1. Let y ∈ Y and v1, . . . , vd ∈ V points over y. Case (1) is a special case of (2), see Morphisms, Lemma 43.10. Case (2) is a special case of (4), see Morphisms, Definition 40.1. If there exists an ample invertible sheaf on V , then there exists an affine open containing v1, . . . , vd by Properties, Lemma 29.5. Thus (3) is true. In cases (4) and (5) it is harmless to replace Y by an affine open neighbourhood of y. Then X is affine too. In case (4) we see that V has an ample invertible sheaf by Morphisms, Definition 37.1 and the result follows from case (3). In case (5) we can replace V by a quasi-compact open containing v1, . . . , vd and we reduce to case (4) by Morphisms, Lemma 38.2. 26. Other chapters

Preliminaries (15) More on Algebra (16) Smoothing Ring Maps (1) Introduction (17) Sheaves of Modules (2) Conventions (18) Modules on Sites (3) Set Theory (19) Injectives (4) Categories (20) Cohomology of Sheaves (5) Topology (21) Cohomology on Sites (6) Sheaves on Spaces (22) Diﬀerential Graded Algebra (7) Sites and Sheaves (23) Divided Power Algebra (8) Stacks (24) Diﬀerential Graded Sheaves (9) Fields (25) Hypercoverings (10) Commutative Algebra Schemes (11) Brauer Groups (12) Homological Algebra (26) Schemes (13) Derived Categories (27) Constructions of Schemes (14) Simplicial Methods (28) Properties of Schemes GROUPOID SCHEMES 53

(29) Morphisms of Schemes (75) Flatness on Algebraic Spaces (30) Cohomology of Schemes (76) Groupoids in Algebraic Spaces (31) Divisors (77) More on Groupoids in Spaces (32) Limits of Schemes (78) Bootstrap (33) Varieties (79) Pushouts of Algebraic Spaces (34) Topologies on Schemes Topics in Geometry (35) Descent (80) Chow Groups of Spaces (36) Derived Categories of Schemes (81) Quotients of Groupoids (37) More on Morphisms (82) More on Cohomology of Spaces (38) More on Flatness (83) Simplicial Spaces (39) Groupoid Schemes (84) Duality for Spaces (40) More on Groupoid Schemes (85) Formal Algebraic Spaces (41) Étale Morphisms of Schemes (86) Algebraization of Formal Spaces Topics in Scheme Theory (87) Resolution of Surfaces Revisited (42) Chow Homology Deformation Theory (43) Intersection Theory (88) Formal Deformation Theory (44) Picard Schemes of Curves (89) Deformation Theory (45) Weil Cohomology Theories (90) The Cotangent Complex (46) Adequate Modules (91) Deformation Problems (47) Dualizing Complexes Algebraic Stacks (48) Duality for Schemes (49) Discriminants and Diﬀerents (92) Algebraic Stacks (50) de Rham Cohomology (93) Examples of Stacks (51) Local Cohomology (94) Sheaves on Algebraic Stacks (52) Algebraic and Formal Geometry (95) Criteria for Representability (53) Algebraic Curves (96) Artin’s Axioms (54) Resolution of Surfaces (97) Quot and Hilbert Spaces (55) Semistable Reduction (98) Properties of Algebraic Stacks (56) Derived Categories of Varieties (99) Morphisms of Algebraic Stacks (57) Fundamental Groups of Schemes (100) Limits of Algebraic Stacks (58) Étale Cohomology (101) Cohomology of Algebraic Stacks (59) Crystalline Cohomology (102) Derived Categories of Stacks (60) Pro-étale Cohomology (103) Introducing Algebraic Stacks (104) More on Morphisms of Stacks (61) More Étale Cohomology (105) The Geometry of Stacks (62) The Trace Formula Topics in Moduli Theory Algebraic Spaces (63) Algebraic Spaces (106) Moduli Stacks (64) Properties of Algebraic Spaces (107) Moduli of Curves (65) Morphisms of Algebraic Spaces Miscellany (66) Decent Algebraic Spaces (108) Examples (67) Cohomology of Algebraic Spaces (109) Exercises (68) Limits of Algebraic Spaces (110) Guide to Literature (69) Divisors on Algebraic Spaces (111) Desirables (70) Algebraic Spaces over Fields (112) Coding Style (71) Topologies on Algebraic Spaces (113) Obsolete (72) Descent and Algebraic Spaces (114) GNU Free Documentation Li- (73) Derived Categories of Spaces cense (74) More on Morphisms of Spaces (115) Auto Generated Index GROUPOID SCHEMES 54

References [ABD+66] Michael Artin, Jean-Etienne Bertin, Michel Demazure, Alexander Grothendieck, Pierre Gabriel, Michel Raynaud, and Jean-Pierre Serre, Schémas en groupes, Sémi- naire de Géométrie Algébrique de l’Institut des Hautes Études Scientiﬁques, Institut des Hautes Études Scientiﬁques, Paris, 1963/1966. [KM97] Sean Keel and Shigefumi Mori, Quotients by groupoids, Ann. of Math. (2) 145 (1997), 193–213. [Mum70] David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Oxford University Press, 1970. [MvdGE] Ben Moonen, Gerard van der Geer, and Bas Edixhoven, Abelian varieties. [Oor66] Frans Oort, Algebraic group schemes in characteristic zero are reduced, Invent. Math. 2 (1966), 79–80. [Per75] Daniel Perrin, Schémas en groupes quasi-compacts sur un corps et groupes henséliens, Publications Mathématiques d’Orsay 165, 75-46 (1975), 148. [Per76] , Approximation des schémas en groupes, quasi compacts sur un corps, Bull. Soc. Math. France 104 (1976), no. 3, 323–335.